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.
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.
In quantum theory, 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). 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
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 (P φ = φ), the other is odd (P φ = –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.
Newton's equation of motion F = m a (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.
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.
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: P ψ(r) = ψ(-r). One must have P2 ψ(r) = ei φ ψ(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 ei φ. If P2 is an element ei Q of a continuous U(1) symmetry group of phase rotations then e-i Q/2 is part of this U(1) and so is also a symmetry. In particular we can define P'=Pe-i Q/2 which is also a symmetry and so we can choose to call P' our parity operator instead of P. Notice that P'2=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.
When 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 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:
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:
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:
There is a straightforward extension of these arguments to scalar field theories which shows that scalars have even parity, since
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 ei Q, it is possible to redefine the parity operator so that P2=1. However, if there exist Majorana neutrinos, which experimentalists today believe is quite likely, 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.
In the 1954 paper Absorption of Negative Pions in Deuterium: Parity of the Pion, William Chinowsky and Jack Steinberger demonstrated that the pion π 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 neutrons n
Neutrons are fermions 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 (-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.
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.
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.
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.
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.
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