Master's Thesis in Theoretical Physics - Universiteit Utrecht

We could also use the conformal FLRW metric by making the substitution dt = a(η)dη, i.e.g µν = a 2 (η)η µν , such that the field equations (5.48) and (5.51) becomeφ ′′0 + 2 a′a φ′ 0 + a2 ξRφ 0 + a 2 λφ 3 0= 0δφ ′′ + 2 a′a δφ′ − ∇ 2 φ + a 2 (ξR + m 2 )δφ = 0. (5.52)Now we perform a conformal rescaling of our fields, φ 0 → ϕ 0 = aφ 0 and δφ → δϕ = aδφ,which allows us to write( )ϕ ′′0 + a 2 ξR − a′′ϕ 0 + λϕ 3 0= 0a)δϕ ′′ +(−∇ 2 + a 2 ξR − a′′a + a2 m 2 δϕ = 0. (5.53)The field ϕ 0 = aφ 0 is a classical field, so we do not need to quantize this field. To quantize thequantum fluctuation δφ = δϕ/a we first calculate the Hamiltonian by using the Lagrangiandensity (5.49) and the conformal FLRW metric∫H = d 3 x [ πδφ ′ − L ′]∫ [ 1= d 3 x2 π2 + a 2 (∇δφ) 2 + 1 ( 2 a4 m 2 + ξR ) ]δφ 2 , (5.54)where the conjugate momentum isˆπ(η,x) =∂L∂∂ η δφ = a2 δ ˆφ′ . (5.55)We quantize the field δφ by imposing the canonical commutator[δ ˆφ(η,x), ˆπ(η,x ′ ) ] = iδ 3 (x − x ′ ), (5.56)The canonical commutator is satisfied if we expand the quantum fluctuation δφ in the followingway∫ dδ ˆφ(η,x) 3 k (â−=(2π) 3 k δφ k(η) + â + −k δφ∗ k (η)) e ik·x , (5.57)andˆπ(η,x) = a 2 ∫d 3 k(2π) 3 (â−k δφ′ k (η) + â+ −k δφ∗′ k (η)) e ik·x . (5.58)Substituting Eq. (5.57) and Eq. (5.58) into the commutator (5.56), we find that the commutationrelation is satisfied when[â−and when the modes satisfy the normalization conditionk , â+ k ′ ]= (2π) 3 δ 3 (k − k ′ ), (5.59)W[δφ k ,δφ ∗ k ] ≡ δφ kδφ ∗′k − δφ∗ k ,δφ′ k = ia 2 , (5.60)where W is the Wronskian between the two modes. Now we rewrite the expansion (5.57)with the conformally rescaled field δϕ = aδφ, such that the mode functions become δϕ k =aδφ k . So we have the expansion∫δ ˆϕ(η,x) =d 3 k(2π) 3 (â−k δϕ k(η) + â + −k δϕ∗ k (η)) e ik·x , (5.61)