Antiunitary Operators Sample Clauses

Antiunitary Operators. An operator T is said to be antilinear, if for any state vectors |ϕ⟩, |ψ⟩ and complex numbers α, β, it satisfies T (α |ϕ⟩ + β |ψ⟩) = α∗T |ϕ⟩ + β∗T |ψ⟩ . (1.37) The asterisk denotes complex conjugation. If in addition T has the prop- erty |⟨ψ|ϕ⟩| = |⟨T ψ|Tϕ⟩|, (1.38) it is called antiunitary [13]. The relations (1.37) and (1.38) lead to the equality16 ⟨Tψ| Tϕ⟩ = ⟨ψ| ϕ⟩∗, (1.39) which can equivalently be taken as the definition of antiunitarity [15]. The operator C of complex conjugation (with respect to the (orthogo- nal) basis {|n⟩}) is an antiunitary operator that satisfies C |n⟩ = |n⟩ ∀n, and C2 = 1. (1.40) 16Note that the use of Dirac bra-ket notation, developed for linear vector spaces, is a risky business when dealing with antilinear operators. The safest approach is to let T first act on a ket, and only then use the dual correspondence to find the corresponding bra [14]. Σ The action of C on a general state vector |ψ⟩ = cn |n⟩ (1.41) n Σ is completely determined by these properties C |ψ⟩ = c∗n |n⟩ . (1.42) n Σ In particular, if |ϕ⟩ = dn |n⟩ (1.43) n Σ we can confirm the antiunitary property (1.39) ⟨Cψ|Cϕ⟩ = cndn∗ = ⟨ψ|ϕ⟩∗. (1.44) n A product of an antiunitary and a unitary operator is again antiu- nitary, while the product of two antiunitary operators is unitary. Every antiunitary operator T can therefore be written as a product of a unitary operator U and the complex conjugation operator C (the form of U will depend on the basis with respect to which C is defined) T = UC. (1.45) In particular, the time reversal operator, our prime example of an antiu- nitary symmetry (and the reason for using here the symbol T to represent an antiunitary operator), will always be written in this form.