Master's Thesis in Theoretical Physics - Universiteit Utrecht

such that the kinetic term transforms as −gg µν (∂ µ φ)(∂ ν φ) = √ −ḡḡ µν( 1(∂ µ ¯φ)(∂ν ¯φ) +4 (D − 2)2 (∂ µ lnΩ)(∂ ν lnΩ)+ D − 2)2 ( ¯φ(∂µ ¯φ)∂ν lnΩ + ¯φ(∂ν ¯φ)∂µ lnΩ)= √ −ḡḡ µν( 1(∂ µ ¯φ)(∂ν ¯φ) +4 (D − 2)2 ¯φ2 (∂ µ lnΩ)(∂ ν lnΩ)+ D − 22 (∂ µ ¯φ)2 )∂ ν lnΩ= √ −ḡḡ µν( 1)(∂ µ ¯φ)(∂ν ¯φ) +4 (D − 2)2 ¯φ2 (∂ µ lnΩ)(∂ ν lnΩ)− D − 2 ¯φ 2 √∂ µ −ḡḡ µν ∂ ν lnΩ2= √ −ḡḡ µν D − 2√ (∂ µ ¯φ)(∂ν ¯φ) [ − −ḡ ¯φ2 ¯□lnΩ2+ 1 ]2 (D − 2)ḡρλ (∂ ρ lnΩ)(∂ λ lnΩ) .In the third step I have partially integrated the last term and dropped the boundary term.In the final step we have used that1 −ḡ∂ ρ√−ḡḡ ρλ ∂ λ = ¯□ = ḡ ρλ ¯ ∇ ρ ¯ ∇ λ . (4.42)Using the new rescaled fields we find that the action is∫S = d D x √ {−ḡ − 1 12 ḡµν (∂ µ ¯φ)(∂ν ¯φ) −2 ξ ¯R ¯φ2 − Ω −D V (Ω D−22 ¯φ) (4.43)( ) [D − 2+ − (D − 1)ξ ¯φ 2 ¯□lnΩ + 1 }42 (D − 2)ḡρλ (∂ ρ lnΩ)(∂ λ lnΩ)].Now we insert a general potential into the actionV (φ) = 1 2 m2 φ 2 + 1 4 λφ4 , (4.44)such that the action is∫S = d D x √ {−ḡ − 1 12 ḡµν (∂ µ ¯φ)(∂ν ¯φ) −2 ξ ¯R ¯φ2 − Ω 2 1 2 m2 ¯φ2 − 1 4 λΩD−4 ¯φ4( ) [D − 2+ − (D − 1)ξ ¯φ 2 ¯□lnΩ + 1 }42 (D − 2)ḡρλ (∂ ρ lnΩ)(∂ λ lnΩ)].(4.45)If ξ now has the special valueξ = D − 24(D − 1) , (4.46)we find that the action is invariant in D ≠ 4 dimensions if m 2 = 0 and λ = 0. However, in4 dimensions we see that the action is invariant under a conformal transformation whenm 2 = 0 and ξ = 1 6 . Thus, the action Eq. (4.1) with a λφ4 potential is conformally invariant ifξ = 1 6 .As an example we will now consider the conformal FLRW metric, see section B.3. The metrichas the form g µν (x) = a 2 (η)η µν , so we can recognize a(t) = Ω(x). Eqs. (B.21), (B.23) and(B.25) can now easily be derived if we appreciate that in a flat Minkowsky space time theChristoffel connection vanishes (in Cartesian coordinates). We have seen that the action in