Master's Thesis in Theoretical Physics - Universiteit Utrecht

This allows us to write the field expansions (5.36) and (5.37) as∫ dˆφ(x, 3 k(t) =(2π) 3 â − k uin k + â+ k (uin k )∗)∫ dˆφ(x, 3 k (t) =ˆb−(2π) 3 k uout k+ ˆb + k (uout k )∗) . (5.39)Note the change of sign of k in the creation operators â + k and ˆb + . Since the combinationsku in and (uink k )∗ , as well as the combinations u out and (uoutkk)∗ form a complete set of solutions,we are allowed to express the in modes in terms of the out modes asu ink = ∫ d 3 k ′(αk(2π) 3 ′ ku outk ′+ β k ′ k(u outk ′ ) ∗) . (5.40)Imposing the canonical commutator (5.9) on the field in Eq. (5.36), we find the followingcondition on the coefficients∫d 3 j()(2π) 3 α kj α ∗ jk − ′ β kjβ ∗ jk= ′ δ kk ′. (5.41)Substituting this expansion into Eq. (5.36), we find that we can express the creation andannihilation operators in the out region as∫ dˆb + 3 k ′ (k=α∗(2π) 3 k ′ kâ+ k+ ′ β k ′ kâ − )k ′∫ dˆb − 3 k ′ (k=αk(2π) 3 ′ kâ − k + ) ′ β∗ k ′ kâ+ k . (5.42)′This is a generalized Bogoliubov transformation as in Eq. (5.32). Consider the followingsituation: initially we are in the in region where we have the vacuum |0 in 〉 with zero particlenumber and energy E 0 . We are now in a position to calculate the particle number in theout region by using the new creation and annihilation operators from Eq. (5.42). For theparticle number, this gives∫〈0 in |N outk |0 in〉 = 〈0 in | ˆb + ˆb d − 3k k |0 k ′in〉 =(2π) 3 |β k ′ k| 2 . (5.43)Thus we see that although the particle number is zero in the in region, it becomes nonzeroin the out region. The important requirement is the time dependence of the frequencyof the harmonic oscillator. This happens for example for a uniform accelerated observerin Minkowski space. Even though the fields are in the zero particle vacuum state, theuniform accelerated observer will detect a distribution of particles similar to a thermalbath of blackbody radiation. This effect is called the Unruh effect.Another example of a scalar field theory with a time dependent frequency is a scalar fieldin an expanding universe. The vacuum can be the zero particle state at some instant oftime, but at a later time we would find particles in this vacuum. Therefore, we can say thatparticles are created by the expansion of the universe. In the next section we will look moreclosely at quantum field theory in an expanding universe. We will quantize our nonminimalscalar inflationary model, and see that we can again write this as a harmonic oscillatorwith a time dependent frequency. The choice of the vacuum state will therefore be difficultbecause there is not a unique choice, but we will see that we can define a natural vacuumstate in quasi de Sitter space, the so-called Bunch-Davies vacuum. With this vacuum choicewe can eventually derive the scalar propagator. In section 5.3 we will generalize our findingsto the two-Higgs model. We will quantize this theory and find the scalar propagator, whichwill be useful in calculating quantum effects of the inflaton on the dynamics of fermions inchapter 6.