Master's Thesis in Theoretical Physics - Universiteit Utrecht

(5.61), we will find a nonzero propagator. The physical propagator is the time orderedFeynman propagator, defined asi∆(x, x ′ ) ≡ 〈0|T[δ ˆϕ(x)δ ˆϕ(x ′ )]|0〉= θ(η − η ′ )〈0|δ ˆϕ(x)δ ˆϕ(x ′ )|0〉 + θ(η ′ − η)〈0|δ ˆϕ(x ′ )δ ˆϕ(x)|0〉, (5.89)where θ is the Heaviside θ-function. The Feynman propagator is Lorentz invariant, andhence the physical propagator. We are actually able to calculate this propagator explicitly ifwe insert the expansion (5.61) with the mode functions (5.83). For our vacuum state we willtake the Bunch-Davies vacuum with α = 1, β = 0. The Feynman propagator in Eq. (5.89)then becomesi∆(x, x ′ ) = π 4{×√ηη′aa ′ ∫ d 3 k(2π) 3 ei⃗ k·(⃗x−⃗x ′ )θ(η − η ′ )H (1)ν (−kη)H(2) ν (−kη′ ) + θ(η ′ − η)H (1)ν (−kη′ )H (2)ν (−kη) }. (5.90)This integration has been performed in [33] in a D-dimensional FLRW universe. The importanceof this becomes clear when we calculate one loop fermion self energy diagrams inchapter 6. We will use dimensional regularization to renormalize the infinite self energy,and therefore we need the scalar and fermion propagators in D dimensions. To calculatethe propagator in D dimensions we need the solutions for the field fluctuations δφ k in Ddimensions, and only a few things will change with respect to the four dimensional case.We can still write the field equation for the fluctuations in the form (5.74), but we haveused a new rescaling for the fields, ϕ = a D 2 −1 φ. Furthermore we can recognize the conformalcoupling factor 1 6, but in D dimensions this isD−24(D−1), see Eq. (4.46). Finally the Ricci scalaris given by Eq. (B.25) in D dimensions. If we now again write the field equation for δϕ k inthe form of Bessel’s equation (5.75), we find that the index ν is nowν 2 D==( ) D − 1 − ɛ 2+2(1 − ɛ)( ) D − 1 − ɛ 2+2(1 − ɛ)1 ξR D + m 2(1 − ɛ) 2 H 2(D − 1)(D − 2ɛ)(1 − ɛ) 2 ξ +1 m 2(1 − ɛ) 2 H 2 . (5.91)In four dimensions we retrieve the ν from Eq. (5.76). We find that the solutions for δϕ kare again Hankel functions, but this time the index is ν D . In D dimensions the scalarpropagator then takes the formi∆(x, x ′ ) = π √ηη′ ∫4 (aa ′ ) D 2{−1×d D−1 k(2π) D−1 ei⃗ k·(⃗x−⃗x ′ )θ(η − η ′ )H (1)ν (−kη)H(2) ν (−kη′ ) + θ(η ′ − η)H (1)ν (−kη′ )H (2)ν (−kη) }. (5.92)Again, in 4 dimensions the mass term precisely cancels against the main contribution comingfrom the ξR term. The integration (5.92) can now be performed to give the propagatorin D dimensions,i∆(x, x ′ ) ∞ = [(1 − ɛ)2 HH ′ ] D 2 −1(4π) D 2× 2 F 1( D − 12Γ( D−12 + ν D)Γ( D−12 − ν D)+ ν D , D − 12Γ( D 2 )− ν D ; D 2 ;1 − y 4), (5.93)