Parity (physics)

In physics, a parity transformation (also called parity) is the simultaneous flip in the sign of all spatial coordinates:

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

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.

Contents

1 Simple symmetry relations

2 Classical mechanics

3 Quantum mechanics

4 Quantum field theory

4.1 Parity violation
4.2 Intrinsic parity of hadrons

5 See also

6 References and external links

Simple symmetry relations

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

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

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

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 (Pφ=φ), the other is odd (Pφ=-φ). These are useful in quantum mechanics.

Classical mechanics

Newton's equation of motion F=ma equates two vectors, and hence is invariant under parity. The law of gravity also involves only vectors and is also, therefore, invariant under parity.

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.

Quantum mechanics

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Parity_1drep.png

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 an unitary operator in quantum mechanics, acting on a wavefunction ψ as follows: Pψ(r) = ψ(-r). Clearly, one must have P²ψ(r)=e^iφψ(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.

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 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(r)=V(-r). The following facts can be easily proven—

If |A> and |B> have opposite parity, then <A|X|B>=0 where X is the position operator.
For a state |L,m> of orbital angular momentum L with z-projection m, P|L,m>=(-1)^{L^|L,m>.}
If [H,P]=0, then a non-degenerate eigenstate of H is an eigenstate of parity.
If [H,P]=0, then no transitions occur between states of opposite parity.

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.

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

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.

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

P a(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.

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 participate in weak interactions. (Notice that this means a right-handed neutrino would not interact by any of the strong, weak or electromagnetic forces.)

The history of the discovery of parity violation is interesting. It was suggested several times, 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.

Intrinsic parity of hadrons

To every particle one can assign an intrinsic parity as long as nature preserves parity. Although weak interactions do not, one can still assign a parity to any hadron by examining the strong interaction reaction that produces it, or through decays not involving the weak interaction, such as π⁰→γγ.

References and external links

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