Quantum Theory - Particle Physics Group

CHAPTER 9. SYMMETRIES AND TRANSFORMATION GROUPS 179While coordinates and momenta are (polar) vectors, i.e. odd under parity, the angular momentum⃗x × ⃗p is a pseudo vector, or axial vector), i.e. even under parityL = ⃗x × ⃗p ⇒ P ⃗ LP † = ⃗ L, P ⃗ S P † = ⃗ S (9.111)Electromagnetic and strong interactions, as well as gravity, preserve parity. The form of theMaxwell equations implies that the electric field is a vector while the magnetic field transformsas an axial vectorP ⃗ E P † = − ⃗ E, P ⃗ B P † = ⃗ B. (9.112)In the relativistic notation A µ → A µ , i.e. A 0 is parity even and the vector potential ⃗ A is parityodd. The spin-orbit coupling ⃗ L ⃗ S and the magnetic coupling ⃗ B( ⃗ L+2 ⃗ S) are axial-axial couplingsand hence allowed by parity, while ⃗ E ⃗ S would be a vector-axial coupling and is hence forbiddenby parity. The parity of the spherical harmonics iswhich is the basis for parity selections rules in atomic physics.P|Y lm 〉 = (−1) l |Y lm 〉 (9.113)Parity is violated in weak interactions, as was first observed in the radioactive β-decay ofpolarized 60 Co. Since spin is parity-even the emission probability has to be the same for theangles θ and π −θ if parity is conserved. But experiments show that most electrons are emittedopposite to the spin direction θ > π/2.Time reversal. For a real Hamiltonian the effect of an inversion of the time direction canbe compensated in the Schrödinger equation by complex conjugation of the wave fuctiont → t ′ = −t ⇒ i ∂∂t ′ψ∗ = i ∂−∂t ψ∗ = Hψ ∗ . (9.114)Time reversal therefore is implemented in quantum mechanics by an anti-unitary operatorT ψ(t,⃗x) = ψ ∗ (−t,⃗x) ⇒ 〈T ϕ|T ψ〉 = 〈ϕ|ψ〉 ∗ , T α|ψ〉 = α ∗ T |ψ〉 (9.115)which implies complex conjugation of scalar products but leaves the norms √ 〈ψ|ψ〉 invariant.For antilinear operators Hermitian conjugation can be defined by 〈ϕ|T † |ψ〉 = 〈ψ|T |ϕ〉. Antiunitaryis then equivalent to antilinearity and T T † = 1. Since velocities and momenta changetheir signs under time inversion we haveT X i T −1 = X i , T P i T −1 = −P i (9.116)The above formulas are compatible with the canonical commutation relationsT [P i ,X j ]T −1 = [−P i ,X j ] = − i δ ij = T i δ ijT −1 . (9.117)