Antiunitary operators and complex conjugation K

Wigner's theorem (topic 1.01, quoted) lets a symmetry keep every overlap ⟨a|b⟩, or keep its modulus and conjugate it. The second kind is antiunitary: Θ(c|a⟩ + d|b⟩) = c*Θ|a⟩ + d*Θ|b⟩ and ⟨Θa|Θb⟩ = ⟨a|b⟩*. Antilinear is not nonlinear. Sums and real multiples pass straight through and only i is reversed, so as operators ΘiΘ⁻¹ = −i and Θ(iħ)Θ⁻¹ = −iħ. Probabilities survive, since |⟨a|b⟩*|² = |⟨a|b⟩|². A one-dimensional space shows the whole structure: an antiunitary is c → e^(iα)c*, a reflection of the complex plane in the line at angle α/2, which no rotation c → e^(iβ)c can imitate. So no continuous family starting at the identity, and no Hamiltonian, can generate Θ, while a product of two antiunitaries is unitary (topic 1.01, quoted). Units throughout are SI with ħ = 1.0546 × 10⁻³⁴ J s kept explicit; nothing here is relativistic, so no metric signature enters.

Pick an orthonormal basis |n⟩, phases included, and let K conjugate components: K Σ cₙ|n⟩ = Σ cₙ*|n⟩. K is antiunitary, because ⟨Ka|Kb⟩ = Σ aₙbₙ* = ⟨a|b⟩*, and K² = 1. Given any antiunitary Θ, set U = ΘK. Then U(c|a⟩) = Θ(c*K|a⟩) = cΘK|a⟩, so U is linear, and ⟨Ua|Ub⟩ = ⟨Ka|Kb⟩* = ⟨a|b⟩, so U is unitary. Hence Θ = UK, derived: on a column of components Θ acts as c → U c*, conjugate first, then multiply. On a spin half, Θ|↑⟩ = |↓⟩ and Θ|↓⟩ = |↑⟩ define Θ = σₓK. Two particles share one K, conjugating product-basis components, and Θ_A ⊗ Θ_B = (U_A ⊗ U_B)K is well defined. K on particle A alone is not: c(|a⟩ ⊗ |b⟩) is both (c|a⟩) ⊗ |b⟩ and |a⟩ ⊗ (c|b⟩), and the two readings would conjugate c differently.

Conjugating components means something only once the basis is named. If |n′⟩ = W|n⟩, conjugation in the new basis is K′ = WKW⁻¹, and since KW⁻¹ = WᵀK, K′ = WWᵀK, derived, with Wᵀ the transpose in the old basis. So K′ = K exactly when W is real. For a spin half with |±y⟩ = (|↑⟩ ± i|↓⟩)/√2, WWᵀ = σ_z and K_y = σ_zK_z. The two disagree about the same ket: K_z|+y⟩ = |−y⟩, while K_y|+y⟩ = |+y⟩. K_z keeps Sₓ and S_z and reverses S_y, the one imaginary Pauli matrix; K_y reverses Sₓ instead. Phases count too: rephasing |n⟩ → e^(iφₙ)|n⟩ multiplies K by diag(e^(2iφₙ)). Every such K squares to 1 and mirrors the Bloch sphere in one plane, so no basis gives a K that reverses all three spin components; topic 1.06 shows why spin's time reversal needs more than K.

With ⟨x|p⟩ = e^(ipx/ħ)/√(2πħ), Kₓ sends ψ(x) to ψ*(x), and conjugating ⟨x|p⟩ gives ⟨x|−p⟩, so Kₓ|p⟩ = |−p⟩, derived. Then Kₓ x Kₓ⁻¹ = x and Kₓ p Kₓ⁻¹ = −p, since −iħ d/dx loses the sign of its i. An electron packet with mean wavenumber 5.0 nm⁻¹, moving at ħk/mₑ = 5.79 × 10⁵ m s⁻¹, becomes one with the same |ψ(x)|² moving at −5.79 × 10⁵ m s⁻¹. Conjugating momentum amplitudes instead gives Kₚψ(x) = ψ*(−x), so Kₚ = ΠKₓ with Π the parity, a unitary. For an oscillator, with x and p in units of √(ħ/mω) and √(ħmω), conjugation in the rotated position kets e^(iθN)|q⟩, eigenkets of x cos θ + p sin θ, is K_θ = e^(2iθN)Kₓ, since the number operator N is real in the position basis: the plane of (⟨x⟩, ⟨p⟩) is mirrored in the line at angle θ. Check: Kₓ[x, p]Kₓ⁻¹ = −iħ = [x, −p], J s on both sides.

For a linear operator ⟨a|A|b⟩ is the same whether A acts right or left. For an antilinear Θ it is not. The map |b⟩ → ⟨a|Θb⟩ is antilinear in |b⟩, so ⟨a|Θ is not a bra at all. The natural attempt to make one, using the antiunitary adjoint fixed by ⟨a|Θb⟩ = ⟨b|Θ†a⟩, with Θ† = Θ⁻¹, gives ⟨Θ†a|b⟩ = ⟨a|Θb⟩*: the complex conjugate of what was meant. Take K conjugating components in the basis |1⟩, |2⟩, with |a⟩ = (3|1⟩ + 4i|2⟩)/5 and |b⟩ = (|1⟩ + |2⟩)/√2. Then ⟨a|Kb⟩ = (3 − 4i)/(5√2) = 0.424 − 0.566i, while ⟨Ka|b⟩ = 0.424 + 0.566i: the same symbols, the opposite phase, with K² = 1 so K† = K. The convention from here on is that Θ acts on the ket to its right, written ⟨a|Θb⟩ or ⟨Θa|Θb⟩, and never as ⟨a|Θ|b⟩ with Θ free to move left.

If states go to Θ|a⟩, operators go to A′ = ΘAΘ⁻¹, which is linear, since its two antilinear factors cancel; numbers inside A are conjugated, and ΘA†Θ⁻¹ = (ΘAΘ⁻¹)†. Now let |γ⟩ = A†|a⟩. Then ⟨a|A|b⟩ = ⟨b|γ⟩* = ⟨Θb|Θγ⟩ = ⟨Θb|ΘA†Θ⁻¹|Θa⟩, derived. For Hermitian A with ΘAΘ⁻¹ = ηA, η = ±1, this is ⟨a|A|b⟩ = η⟨Θb|A|Θa⟩: bra and ket exchange, and odd operators change sign. Check with K_z on a spin half: ⟨↑|S_y|↓⟩ = −iħ/2, S_y is odd, and −⟨↓|S_y|↑⟩ = −iħ/2. Two limits. At a = b, ⟨A⟩ in Θ|a⟩ is η⟨A⟩ in |a⟩, so an odd A has zero mean in any state Θ returns up to a phase. For a unitary U the same steps give ⟨Ua|UAU⁻¹|Ub⟩ with no exchange. The exchange is the fingerprint of antiunitarity, and Kramers, the dipole theorem and reciprocity (topics 1.07 to 1.09) apply this line.