Permanent electric dipole moments violate both P and T

Put a system in a weak uniform field and expand a level's energy W in the field E: W = W₀ − ⟨d⟩⋅E − ½αE² + … . A moment is permanent when the linear coefficient survives as E → 0, that is, when a stationary state of the isolated system has ⟨d⟩ ≠ 0 with no field applied; an induced moment, αE, vanishes with the field. The lesson works in SI units, quoting moments in the e cm the experiments use (1 e cm = 1.602 × 10⁻²¹ C m); nothing here is relativistic, so the course's metric (+, −, −, −) never enters. For scale, e a₀ = 8.48 × 10⁻³⁰ C m = 5.29 × 10⁻⁹ e cm, and the best electron bound, 4.1 × 10⁻³⁰ e cm, is twenty-one orders of magnitude below it.

Let the level be a single multiplet |α j m⟩, m = −j … j, degenerate with no other state. Quantum Mechanics II's projection theorem, quoted, says every vector operator restricted to it is a multiple of J: ⟨α j m′|d|α j m⟩ = c⟨j m′|J|j m⟩ with c = ⟨J⋅d⟩/(ħ²j(j + 1)). J⋅d is Hermitian, so c is real, and the stretched state m = j carries ⟨d_z⟩ = cjħ, the number called the moment. For j = 0 there is nothing to point along and ⟨d⟩ = 0 by rotation invariance alone, the trivial limit. The magnetic moment obeys the same theorem, μ = γJ, and is allowed: μ and J are both axial and both reverse under T, so γ is even under both symmetries. The dipole is polar and T-even, the opposite of J on both counts.

Suppose [P, H] = 0. P also commutes with rotations, so it maps the isolated multiplet onto itself and commutes with J_z and with the ladder operators; each |α j m⟩ is then a parity eigenstate with one common eigenvalue η = ±1. Since P d P⁻¹ = −d and P J P⁻¹ = +J, conjugating d = cJ by P gives −d = cJ, so c = −c and c = 0, derived. Directly: ⟨d⟩ = ⟨ψ|P⁻¹(−d)P|ψ⟩ = −η²⟨d⟩. Parity alone is a weak protection, though. The weak interaction breaks it, measured since 1957, and atomic parity violation has been seen in caesium. Yet a moment needs more than broken P, because time reversal forbids it independently.

Suppose [Θ, H] = 0. Then Θ maps the multiplet onto itself, and because Θ J_z Θ⁻¹ = −J_z, it sends |α j m⟩ to a phase times |α j −m⟩ (topic 1.06). For an antiunitary Θ, ⟨Θψ|A|Θψ⟩ = ⟨ψ|Θ⁻¹AΘ|ψ⟩*, and d_z is Hermitian and T-even, so ⟨d_z⟩ in the state −m equals ⟨d_z⟩ in m: T makes the moment even in m. The projection theorem makes it cmħ, odd in m. At m = j ≠ 0 both hold only if c = 0, derived. The phase cancels, so Θ² = ±1 never enters. With either symmetry enough, a nonzero moment needs P and T both broken. By the CPT theorem, quoted here, it then breaks CP too, which is why these searches probe new sources of CP violation.

In the Coulomb model hydrogen's 2s and 2p are degenerate, so n = 2 is not one j multiplet. A field mixes them into (|2s⟩ ± |2p₀⟩)/√2, shifted by ±3 e a₀ E (Quantum Mechanics II): linear, yet those states are not parity eigenstates and carry ⟨L⟩ = 0, so their dipole is not tied to J, neither proof applies, and H stays P- and T-invariant. The Lamb shift, 1057.8 MHz (quoted), restores isolation: with w = √3 e a₀ the 2s₁/₂ shift stays quadratic below Δ/2w ≈ 239 V cm⁻¹. Quantum Mechanics I's ammonia is the same two-level problem, Δ/h = 23.79 GHz and w = 1.47 D = 0.578 e a₀, so Δ/2w ≈ 16 kV cm⁻¹. A polar molecule's 1.85 D, water's say, is a body-frame number: a free molecule's stationary states have definite parity, and a lab field merely mixes close opposite-parity rotational levels. An ion at a crystal site with no inversion centre is the third escape, with rotation invariance itself gone.

For an electron H = −dₑ σ⋅E_eff, but a bound non-relativistic electron feels zero average field, since ⟨F⟩ = 0 in a stationary state: Schiff's theorem, quoted. The signal is relativistic and grows steeply with nuclear charge, so experiments use heavy polar molecules, fully polarised by fields of order 100 V cm⁻¹ because their parity doublets are so close. The electron then feels a calculated E_eff of about 78 GV cm⁻¹ in ThO and 23 GV cm⁻¹ in HfF⁺ (quoted). The measured bounds on |dₑ|, at 90% confidence, are 1.1 × 10⁻²⁹ e cm from ThO and 4.1 × 10⁻³⁰ e cm from HfF⁺ (quoted). At the HfF⁺ bound, dₑE_eff = 9.4 × 10⁻²⁰ eV, and the splitting 2dₑE_eff is h × 4.6 × 10⁻⁵ Hz. The neutron's bound, 1.8 × 10⁻²⁶ e cm (quoted), keeps any charge separation along its spin below 2 × 10⁻²⁶ cm, two parts in 10¹³ of its own size. Every measurement so far is consistent with zero.