Master's Thesis in Theoretical Physics - Universiteit Utrecht

great triumphs of inflation is that it predicts a nearly scale invariant spectrum of densityfluctuations. Indeed, the WMAP mission has observed the spectrum of density perturbations,and the result is that the spectrum is almost scale invariant, but not quite. Thisnear-scale invariance is a generic feature of inflation, but the actual size of the deviationsof a scale invariant spectrum is model dependent. The deviations from scale invariance areproportional to the slow-roll parameters, which thus constrain potentials, but do not give aspecific form of the potential.We now return to our nonminimal inflationary model. Our goal in this chapter is to successfullyquantize the nonminimal inflationary model. We will apply this in chapter ??to calculate one-loop radiative corrections to the fermion propagator. However, as we willsee in the next sections, quantizing a field theory is straightforward in Minkowski space,but quite subtle in an expanding universe. For good literature on quantum field theory incurved backgrounds, see [30] and [31]. One of the main problems is that the vacuum atone point in time is not the same vacuum at some later time. This leads to the very specialconsequence that particles can be created through the expansion of the universe.The outline of this chapter will be the following. In section 5.1 we will quickly introducequantum field theory in Minkowski space. We will quantize our theory by introducing creationand annihilation operators, define a vacuum and show that this remains the state oflowest energy by a specific choice of quantizing our fields. In section 5.2 we will quantizethe nonminimally coupled inflaton field and we will see that there is no unique choice forthe vacuum in an expanding universe. However, in an inflationary universe we can makea natural vacuum choice in certain regimes which allows us to quantize the field uniquelyand calculate the scalar propagator. Finally in section 5.3 we will extend our findings andquantize the two-Higgs doublet model. We will see that we can again solve the equations ofmotion exactly which allows us to calculate the propagator for the two-Higgs doublet modelin quasi de Sitter space.5.1 Quantum field theory in Minkowski spaceWe start with a simple action with one real scalar field with a mass m,∫S = d 4 x(− 1 2 ηµν ∂ µ φ∂ ν φ − 1 )2 m2 φ 2∫ ( 1= d 3 xdt2 ˙φ 2 − 1 2 (∇φ)2 − 1 )2 m2 φ 2 . (5.1)Minimizing this action gives the equation of motion for the scalar field φ, which in this caseis the Klein-Gordon equation,0 = −η µν ∂ µ ∂ ν φ + m 2 φ= ¨φ − ∇ 2 φ + m 2 φ. (5.2)To simplify this equation we can perform a Fourier expansion of the field, i.e.∫φ(x, t) =d 3 k(2π) 3 φ k(t)e ik·x , (5.3)where k ≡ |k|. If the field φ is real, we have the additional condition that φ ∗ k = φ −k. Substitutingthis expansion in our action (5.1), we find∫ (S = dt d3 k 1(2π) 3 2 | ˙φk | 2 − 1 )2 (k2 + m 2 )|φ k | 2 , (5.4)