Master's Thesis in Theoretical Physics - Universiteit Utrecht

(5.28) with α = 1 and β = 0, and it remains the lowest energy state if our system evolves intime.The situation changes of course when our Hamiltonian in Eq. (5.21) is nót time independent.For example, the frequency of the harmonic oscillator could be time dependentω k = ω k (t). As we will see in the section 5.2, this is precisely what happens in an expandinguniverse. As we have shown in this section, a harmonic oscillator with a constant frequencyallows us to define a unique vacuum state that is the state of lowest energy at all times. Ifthe frequency is on the other hand time dependent, such a unique vacuum does not exist.We have also seen that the notion of particles is not well defined in these cases. Thereforein an expanding universe, we will abandon the whole particle concept and only work withthe propagator, which is still well-defined. Before we do this however, let us consider a toymodel of a harmonic oscillator with a varying frequency.5.1.2 In and out regionsSuppose we have a harmonic oscillator with a varying frequency, i.e.¨ φ k + ω 2 k (t)φ k = 0. (5.35)In principle we have a different solution for the fields φ k at each instant of time. This willgive us a different expansion in creation and annihilation operators every time, and ourvacuum will be different at each instant of time. So it seems this is as far as we can go.We can make our lives easier however by defining the so-called in and out regions. In theseregions the frequency is approximately constant and we can find solutions of Eq. (5.35). Asa preview to the next section, in an expanding universe we can consider regions where theuniverse is asymptotically flat, i.e. the scale factor becomes constant in these regions.Suppose now that we have some constant frequency ω inkfor t < t 0 (the in region) and aconstant frequency ω out for t > tk1 (the out region). We can now write our field expansion inthe in region asˆφ(x, t) =∫d 3 k((2π) 3 â − k uin k + â+ −k (uin k )∗) e ik·x , (5.36)where the fields u in are the fundamental solutions of Eq. (5.35) as in Eq. (5.26) withkfrequency ω k = ω ink . As we have seen, with this expansion the vacuum defined by â− k |0 in〉 = 0is also the state of lowest energy. We could do the same for the out region, and writeˆφ(x, t) =∫d 3 k(2π) 3 ( ˆb−k uout k+ ˆb + −k (uout k )∗) e ik·x . (5.37)Again, the annihilation operators in the out region also define a vacuum by ˆb − k |0 out〉 = 0,which is the state of lowest energy in the out region. We are now interested in the energydifference between the in and out vacua. The energy difference will basically tell us thatparticles have been created. To relate the vacua, we first have to express u in in terms ofu outk. First we define u ink= u ink eik·x =u outk= u outke ik·x =1√√2ω ink12ω outke i(k·x−ωin k t) ,(u ink )∗ke i(k·x−ωout k t) , (u outk )∗ . (5.38)