Master's Thesis in Theoretical Physics - Universiteit Utrecht

The functions Φ (1)kand in 4 dimensions it isand Φ(2)kare 2×2-matrices because the index ν is a constant 2×2-matrix,ν 2 =( ) 3 − ɛ 2M 2 6(2 − ɛ)Ξ−2(1 − ɛ) H 2 −(1 − ɛ)2(1 − ɛ) 2 .Because the nonminimal coupling and mass matrices are hermitean, the index ν is alsohermitean, ν † = ν. This useful property allows us to derive the important relation( )Φ (1) †k (η) = Φ(2)(η) (5.125)kThe normalization of the mode functions Φ (1) and Φ(2) has been chosen such that the Wronskianof the two solutions satisfiesk k[]W Φ (1)k (η),Φ(2) k (η) = Φ (1) (η) · Φ(2)′ (η) − Φ(2) (η) · Φ(1)′k k k k(η)i=a 2 . (5.126)The canonical commutation relation Eq. (5.116) is satisfied only when condition Eq. (5.121)is fulfilled. Using the Wronskian between the two fundamental solutions from Eq. (5.126)we get the following conditions on the matricesα · α † − γ · γ † = 1β · β † − δ · δ † = −1α · β † − γ · δ † = 0, (5.127)where 1 denotes the 2 × 2 unit matrix. A particular solution for the matrices is α = δ =diag(1,1) and β = γ = 0. This corresponds to an initial vacuum state with zero particles asη → −∞, our familiar Bunch-Davies vacuum. In this special case the two general solutionsare simply Φ k = Φ (1) and Ψ ∗ kk = Φ(2) , such that the Wronskian between the two solutions iskW(Φ k ,Ψ ∗ k ) = i/a2 .In the Bunch-Davies vacuum we can again calculate the propagator for the two-Higgs doubletmodel. There will now be four propagators because we have two fields that can propagateinto each other. To be exact, the time ordered/Feynman propagator is defined asi∆(x, x ′ ) ≡ 〈0|T[Φ(x)Φ † (x ′ )|0〉= 〈0|θ(t − t ′ )Φ(x)Φ † (x ′ ) + θ(t ′ − t)Φ(x ′ )Φ † (x)|0〉. (5.128)Note that we use terms like ΦΦ † to make sure that the propagator is actually a hermitean2 × 2-matrix. To be more precise(i∆(x, x ′ T[φ1 (x)φ ∗) ≡ 〈0|1 (x′ )] T[φ 1 (x)φ ∗ )2 (x′ )]T[φ 2 (x)φ ∗ 1 (x′ )] T[φ 2 (x)φ ∗ |0〉2 (x′ )](= 〈0|θ(t − t ′ φ1 (x)φ ∗)1 (x′ ) φ 1 (x)φ ∗ )2 (x′ )φ 2 (x)φ ∗ 1 (x′ ) φ 2 (x)φ ∗ |0〉2 (x′ )(+〈0|θ(t ′ φ1 (x ′ )φ ∗− t)1 (x) φ 1(x ′ )φ ∗ 2 (x) )φ 2 (x ′ )φ ∗ 1 (x) φ 2(x ′ )φ ∗ 2 (x) |0〉. (5.129)Choosing the Bunch-Davies vacuum with α = δ = 1 and β = γ = 0 we find that the Feynmanpropagator isi∆(x, x ′ ) = π √ηη′ ∫ d 3 keik·(x−x′ )4 aa ′ (2π){3× θ(η − η ′ )H (1)ν (−kη)H(2) ν (−kη′ ) + θ(η ′ − η)H (1)ν (−kη′ )H (2)ν}. (−kη)