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Parity (physics)
Encyclopedia
In physics
, a parity transformation (also called parity inversion) is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:
A 3×3 matrix representation of P would have determinant
equal to −1, and hence cannot reduce to a rotation
which has a determinant equal to 1. The corresponding mathematical notion is that of a point reflection
.
In a two-dimensional plane, parity is not a simultaneous flip of all coordinates, which would be the same as a rotation
by 180 degrees. It is important that the determinant of the P matrix be −1, which does not happen for 180 degree rotation in 2-D where a parity transformation flips the sign of either x or y, not both.
s, classical geometrical objects can be classified into scalars
, vectors, and tensor
s of higher rank. In classical physics
, physical configurations need to transform under representation
s 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 representation
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 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). Projective representations of the rotation group that are not representations are called spinor
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
One can define reflections such as
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
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 (Pφ = φ), the other is odd (Pφ = −φ). 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
.
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.
, the time
when an event occurs
, the mass
of a particle
, the energy
of the particle
, power
(rate of work done)
, the electric charge density
, the electric potential
(volt
age)
, energy density
of the electromagnetic field
, the angular momentum
of a particle (both orbital and spin
) (axial vector)
, the magnetic field
(axial vector)
, the auxiliary magnetic field
, the magnetization
Maxwell stress tensor
, the helicity
, the magnetic flux
, the position of a particle in three-space
, the velocity
of a particle
, the acceleration
of the particle
, the linear momentum of a particle
, the force exerted on a particle
, the electric current density
, the electric field
, the electric displacement field
, the electric polarization
, the electromagnetic vector potential
, Poynting vector
In quantum mechanics
, spacetime transformations act on quantum states. The parity transformation, P, is a unitary operator
in quantum mechanics, acting on a state ψ as follows: Pψ(r) = ψ(−r). One must have P2ψ(r) = eiφψ(r), since an overall phase is unobservable.
The operator P2, which reverses the parity of a state twice, leaves the spacetime invariant and so is an internal symmetry which rotates its eigenstates by phases eiφ. If P2 is an element eiQ of a continuous U(1) symmetry group of phase rotations then e−iQ/2 is part of this U(1) and so is also a symmetry. In particular we can define P = Pe−iQ/2 which is also a symmetry and so we can choose to call P our parity operator instead of P. Notice that P2 = 1 and so P has eigenvalues ±1. However when no such symmetry group exists, it may be that all parity transformations have some eigenvalues which are phases other than ±1.
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 quantum mechanics, Hamiltonian
s 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, i.e., V = V(r), hence the potential is spherically symmetric. The following facts can be easily proven:
Some of the non-degenerate eigenfunctions of H are unaffected (invariant) by parity P and the others will be merely reversed in sign when the Hamiltonian operator and the parity operator commute
:
where c is a constant, the eigenvalue of P,
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.
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:
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 boson
s 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
This is true even for a complex scalar field. (Details of spinor
s are dealt with in the article on the Dirac equation
, where it is shown that fermion
s and antifermions have opposite intrinsic parity.)
With fermion
s, there is a slight complication because there is more than one spin group.
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
neutrino
s, which experimentalists today believe is quite 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.
demonstrated that the pion
has negative parity. They studied the decay of an "atom" made from a deuterium
nucleus (d) and a negatively charged pion (π–) in a state with zero orbital angular momentum
L = 0 into two neutron
s (n).
Neutrons are composed of fermion
s and so obey Fermi 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 L = 1. The total parity is the product of the intrinsic parities of the particles and the extrinsic parity of the spherical harmonic function(−1)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 forementioned 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 deuterium, (−1)(1)2/(1)2, which is equal to minus one. Thus they concluded that the pion is a pseudoscalar particle.
, 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.
It was suggested several times and in different contexts that parity might not be conserved, but in the absence of compelling evidence these suggestions 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.
In 1957 C. S. 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 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 4th 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. 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. The discovery of parity violation immediately explained the outstanding τ–θ puzzle in the physics of kaon
s.
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 Donner Professor of Physics as part of the STAR experiment, which has been smashing atoms together since 2000, showed a variation in the law of parity itself.
s 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.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, a parity transformation (also called parity inversion) is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:
A 3×3 matrix representation of P would have determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
equal to −1, and hence cannot reduce to a rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
which has a determinant equal to 1. The corresponding mathematical notion is that of a point reflection
Point reflection
In geometry, a point reflection or inversion in a point is a type of isometry of Euclidean space...
.
In a two-dimensional plane, parity is not a simultaneous flip of all coordinates, which would be the same as a rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
by 180 degrees. It is important that the determinant of the P matrix be −1, which does not happen for 180 degree rotation in 2-D where a parity transformation flips the sign of either x or y, not both.
Simple symmetry relations
Under rotationRotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
s, classical geometrical objects can be classified into scalars
Scalar (physics)
In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . This is in contrast to a vector...
, vectors, and tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
s of higher rank. In classical physics
Classical physics
What "classical physics" refers to depends on the context. When discussing special relativity, it refers to the Newtonian physics which preceded relativity, i.e. the branches of physics based on principles developed before the rise of relativity and quantum mechanics...
, physical configurations need to transform under representation
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
s of every symmetry group.
Quantum theory
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
predicts that states in a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
do not need to transform under representations of the group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
of rotations, but only under projective representation
Projective representation
In the mathematical field of representation theory, 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 groupwhere GL is the general linear group of invertible linear transformations of V over F and F* here is the...
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 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
Special unitary group
The special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...
SU(2). Projective representations of the rotation group that are not representations are called spinor
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
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 (P = 1) and pseudoscalars (P = −1) which are rotationally invariant.
- vectors (P = −1) and axial vectors (also called pseudovectorPseudovectorIn physics and mathematics, a pseudovector is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation such as a reflection. Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image...
s) (P = 1) which both transform as vectors under rotation.
One can define reflections such as
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
Abelian group
In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...
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 (Pφ = φ), the other is odd (Pφ = −φ). These are useful in quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
. 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 (waves)
Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...
.
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 L 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.
Even
Classical variables, predominantly scalar quantities, which do not change upon spatial inversion include:, the timeTime
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
when an event occurs
, the massMass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
of a particle
, the energyEnergy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
of the particle
, powerPower (physics)
In physics, power is the rate at which energy is transferred, used, or transformed. For example, the rate at which a light bulb transforms electrical energy into heat and light is measured in watts—the more wattage, the more power, or equivalently the more electrical energy is used per unit...
(rate of work done)
, the electric charge densityCharge density
The linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter , square meter , or cubic meter , respectively, and represented by the lowercase Greek letter Rho . Since there are positive as well as...
, the electric potentialElectric potential
In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...
(volt
Volt
The volt is the SI derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honor of the Italian physicist Alessandro Volta , who invented the voltaic pile, possibly the first chemical battery.- Definition :A single volt is defined as the...
age)
, energy densityEnergy density
Energy density is a term used for the amount of energy stored in a given system or region of space per unit volume. Often only the useful or extractable energy is quantified, which is to say that chemically inaccessible energy such as rest mass energy is ignored...
of the electromagnetic field
Electromagnetic field
An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...
, the angular momentumAngular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
of a particle (both orbital and spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
) (axial vector)
, the magnetic fieldMagnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
(axial vector)
, the auxiliary magnetic field, the magnetizationMagnetization
In classical electromagnetism, magnetization or magnetic polarization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material...
Maxwell stress tensorMaxwell stress tensor
The Maxwell Stress Tensor is a mathematical object in physics, more precisely it is a second rank tensor used in classical electromagnetism to represent the interaction between electric/magnetic forces and mechanical momentum...
- 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:, the helicityHelicity
The term helicity has several meanings. In physics, all referring to a phenomenon that resembles a helix. See:*helicity , the extent to which corkscrew-like motion occurs...
, the magnetic fluxMagnetic flux
Magnetic flux , is a measure of the amount of magnetic B field passing through a given surface . The SI unit of magnetic flux is the weber...
, the position of a particle in three-space, the velocityVelocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
of a particle
, the accelerationAcceleration
In physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. ...
of the particle
, the linear momentum of a particle, the force exerted on a particle, the electric current densityCurrent density
Current density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved...
, the electric fieldElectric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
, the electric displacement field, the electric polarization, the electromagnetic vector potentialVector potential
In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose negative gradient is a given vector field....
, Poynting vectorPoynting vector
In physics, the Poynting vector can be thought of as representing the directional energy flux density of an electromagnetic field. It is named after its inventor John Henry Poynting. Oliver Heaviside and Nikolay Umov independently co-invented the Poynting vector...
Possible eigenvalues
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, spacetime transformations act on quantum states. The parity transformation, P, is a unitary operator
Unitary operator
In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfyingU^*U=UU^*=I...
in quantum mechanics, acting on a state ψ as follows: Pψ(r) = ψ(−r). One must have P2ψ(r) = eiφψ(r), since an overall phase is unobservable.
The operator P2, which reverses the parity of a state twice, leaves the spacetime invariant and so is an internal symmetry which rotates its eigenstates by phases eiφ. If P2 is an element eiQ of a continuous U(1) symmetry group of phase rotations then e−iQ/2 is part of this U(1) and so is also a symmetry. In particular we can define P = Pe−iQ/2 which is also a symmetry and so we can choose to call P our parity operator instead of P. Notice that P2 = 1 and so P has eigenvalues ±1. However when no such symmetry group exists, it may be that all parity transformations have some eigenvalues which are phases other than ±1.
Consequences of parity symmetry
When parity generates the Abelian groupAbelian group
In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...
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 quantum mechanics, Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
s are invariant
Invariant (physics)
In mathematics and theoretical physics, an invariant is a property of a system which remains unchanged under some transformation.-Examples:In the current era, the immobility of polaris under the diurnal motion of the celestial sphere is a classical illustration of physical invariance.Another...
(symmetric) under a parity transformation if P commutes
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
with the Hamiltonian
Hamiltonian
Hamiltonian may refer toIn mathematics :* Hamiltonian system* Hamiltonian path, in graph theory** Hamiltonian cycle, a special case of a Hamiltonian path* Hamiltonian group, in group theory* Hamiltonian...
. In non-relativistic quantum mechanics, this happens for any potential which is scalar, i.e., V = V(r), hence the potential is spherically symmetric. The following facts can be easily proven:
- If |A> and |B> have the same parity, then Position operatorIn quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle. Consider, for example, the case of a spinless particle moving on a line. The state space for such a particle is L2, the Hilbert space of complex-valued and square-integrable ...
. - For a state |L, Lz> of orbital angular momentum L with z-axis projection Lz, P|L, Lz> = (−1)L|L, Lz>.
- If [H, P] = 0, then atomic dipole transitions only 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 reversed in sign when the Hamiltonian operator and the parity operator commute
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
:
-
- PΨ = cΨ,
where c is a constant, the eigenvalue of P,
-
- P2Ψ = cPΨ.
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
Vacuum state
In quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles...
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.
To show that quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics 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 special relativity is achieved...
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
Canonical quantization
In physics, canonical quantization is a procedure for quantizing a classical theory while attempting to preserve the formal structure of the classical theory, to the extent possible. Historically, this was Werner Heisenberg's route to obtaining quantum mechanics...
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
Intrinsic parity
In quantum mechanics, the intrinsic parity is a phase factor that arises as an eigenvalue of the parity operation...
. Similarly all vector boson
Vector boson
In particle physics, a vector boson is a boson with the spin quantum number equal to 1.The vector bosons considered to be elementary particles in the Standard Model are the gauge bosons or, the force carriers of fundamental interactions: the photon of electromagnetism, the W and Z bosons of the...
s can be shown to have odd intrinsic parity, and all axial-vectors
Pseudovector meson
In high energy physics, a pseudovector meson or axial vector meson is a meson with total spin 1 and even parity . Compare to a vector meson, which has a total spin 1 and odd parity....
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 spinor
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
s are dealt with in the article on the Dirac equation
Dirac equation
The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
, where it is shown that fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
s and antifermions have opposite intrinsic parity.)
With fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
s, there is a slight complication because there is more than one spin group.
Fixing the global symmetries
In the Standard ModelStandard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
of fundamental interactions there are precisely three global internal U(1) symmetry groups available, with charges equal to the baryon
Baryon
A baryon is a composite particle made up of three quarks . Baryons and mesons belong to the hadron family, which are the quark-based particles...
number B, the lepton
Lepton
A lepton is an elementary particle and a fundamental constituent of matter. The best known of all leptons is the electron which governs nearly all of chemistry as it is found in atoms and is directly tied to all chemical properties. Two main classes of leptons exist: charged leptons , and neutral...
number L and the electric charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
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
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....
, the neutron
Neutron
The neutron is a subatomic hadron particle which has the symbol or , no net electric charge and a mass slightly larger than that of a proton. With the exception of hydrogen, nuclei of atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of...
and the electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
, to be +1.
Steven Weinberg
Steven Weinberg
Steven Weinberg is an American theoretical physicist and Nobel laureate in Physics for his contributions with Abdus Salam and Sheldon Glashow to the unification of the weak force and electromagnetic interaction between elementary particles....
has shown that if P2 = (−1)F, where F is the fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
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
Majorana fermion
In physics, a Majorana fermion is a fermion which is its own anti-particle. The term is used in opposition to Dirac fermion, which describes particles that differ from their antiparticles...
neutrino
Neutrino
A neutrino is an electrically neutral, weakly interacting elementary subatomic particle with a half-integer spin, chirality and a disputed but small non-zero mass. It is able to pass through ordinary matter almost unaffected...
s, which experimentalists today believe is quite 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 SteinbergerJack Steinberger
Jack Steinberger is a German-American physicist currently residing near Geneva, Switzerland. He co-discovered the muon neutrino, along with Leon Lederman and Melvin Schwartz, for which they were given the 1988 Nobel Prize in Physics.-Life:...
demonstrated that the pion
Pion
In particle physics, a pion is any of three subatomic particles: , , and . Pions are the lightest mesons and they play an important role in explaining the low-energy properties of the strong nuclear force....
has negative parity. They studied the decay of an "atom" made from a deuterium
Deuterium
Deuterium, also called heavy hydrogen, is one of two stable isotopes of hydrogen. It has a natural abundance in Earth's oceans of about one atom in of hydrogen . Deuterium accounts for approximately 0.0156% of all naturally occurring hydrogen in Earth's oceans, while the most common isotope ...
nucleus (d) and a negatively charged pion (π–) in a state with zero orbital angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
L = 0 into two neutron
Neutron
The neutron is a subatomic hadron particle which has the symbol or , no net electric charge and a mass slightly larger than that of a proton. With the exception of hydrogen, nuclei of atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of...
s (n).
Neutrons are composed of fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
s and so obey Fermi 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 L = 1. The total parity is the product of the intrinsic parities of the particles and the extrinsic parity of the spherical harmonic function(−1)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 forementioned 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 deuterium, (−1)(1)2/(1)2, which is equal to minus one. Thus they concluded that the pion is a pseudoscalar particle.
Parity violation
Although parity is conserved in electromagnetismElectromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
, 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
Chirality (physics)
A chiral phenomenon is one that is not identical to its mirror image . The spin of a particle may be used to define a handedness for that particle. A symmetry transformation between the two is called parity...
gauge interaction. Only the left-handed components of particles and right-handed components of antiparticles participate in weak interactions in the Standard Model
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
. This implies that parity is not a symmetry of our universe, unless a hidden mirror sector
Mirror matter
In physics, mirror matter, also called shadow matter or Alice matter, is a hypothetical counterpart to ordinary matter.Modern physics deals with three basic types of spatial symmetry: reflection, rotation and translation. The known elementary particles respect rotation and translation symmetry but...
exists in which parity is violated in the opposite way.
It was suggested several times and in different contexts that parity might not be conserved, but in the absence of compelling evidence these suggestions 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
Weak interaction
Weak interaction , is one of the four fundamental forces of nature, alongside the strong nuclear force, electromagnetism, and gravity. It is responsible for the radioactive decay of subatomic particles and initiates the process known as hydrogen fusion in stars...
. They proposed several possible direct experimental tests. They were almost ignored, but Lee was able to convince his Columbia colleague Chien-Shiung Wu
Chien-Shiung Wu
Chien-Shiung Wu was a Chinese-American physicist with expertise in the techniques of experimental physics and radioactivity. Wu worked on the Manhattan Project...
to try it. She needed special cryogenic facilities and expertise, so the experiment was done at the National Bureau of Standards.
In 1957 C. S. 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
Cobalt-60
Cobalt-60, , is a synthetic radioactive isotope of cobalt. Due to its half-life of 5.27 years, is not found in nature. It is produced artificially by neutron activation of . decays by beta decay to the stable isotope nickel-60...
. 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 4th January 1957 at a "Friday Lunch" gathering of the Physics Department of Columbia. Three of them, R. L. Garwin
Richard Garwin
Richard Lawrence Garwin , is an American physicist. He received his bachelor's degree from the Case Institute of Technology in 1947 and obtained his Doctor of Philosophy from the University of Chicago in 1949, where he worked in the lab of Enrico Fermi.Garwin is IBM Fellow Emeritus at the Thomas J...
, Leon Lederman, and R. 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.
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. The discovery of parity violation immediately explained the outstanding τ–θ puzzle in the physics of kaon
Kaon
In particle physics, a kaon is any one of a group of four mesons distinguished by the fact that they carry a quantum number called strangeness...
s.
In 2010, it was reported that physicists working with the Relativistic Heavy Ion Collider
Relativistic Heavy Ion Collider
The Relativistic Heavy Ion Collider is one of two existing heavy-ion colliders, and the only spin-polarized proton collider in the world. It is located at Brookhaven National Laboratory in Upton, New York and operated by an international team of researchers...
(RHIC) had created a short-lived parity symmetry-breaking bubble in quark-gluon plasmas. An experiment conducted by several physicists including Yale's Donner Professor of Physics as part of the STAR experiment, which has been smashing atoms together since 2000, showed a variation in the law of parity itself.
Intrinsic parity of hadrons
To every particle one can assign an intrinsic parity as long as nature preserves parity. Although weak interactionWeak interaction
Weak interaction , is one of the four fundamental forces of nature, alongside the strong nuclear force, electromagnetism, and gravity. It is responsible for the radioactive decay of subatomic particles and initiates the process known as hydrogen fusion in stars...
s do not, one can still assign a parity to any hadron
Hadron
In particle physics, a hadron is a composite particle made of quarks held together by the strong force...
by examining the strong interaction
Strong interaction
In particle physics, the strong interaction is one of the four fundamental interactions of nature, the others being electromagnetism, the weak interaction and gravitation. As with the other fundamental interactions, it is a non-contact force...
reaction that produces it, or through decays not involving the weak interaction
Weak interaction
Weak interaction , is one of the four fundamental forces of nature, alongside the strong nuclear force, electromagnetism, and gravity. It is responsible for the radioactive decay of subatomic particles and initiates the process known as hydrogen fusion in stars...
, such as rho meson decay to pions.