115 Using and 〈 ˆ a + $_{k}$ˆ a − k ′ 〉 ≡ 〈 n k ∣ ˆ a + $_{k}$ˆ a − k ′ ∣ n k 〉 〈 n k | n k 〉 = n k δ ( k − $_{k}$′$_{)}$ (A.4) 〈 ˆ a + $_{k}$ˆ a + k ′ 〉 = 〈 ˆ a − $_{k}$ˆ a − k ′ 〉 = 0 (A.5) [ ˆ $_{a}$− $_{k}$, ˆ a + k ′ ] = δ ( k − $_{k}$′$_{)}$ (A.6) 〈 φ 2 〉 = 1 16 π 3 ∫∫ 1 √ ω$_{k}$ω$_{k}$ ′ ( e − iω$_{k}$t + i k · x + iω$_{k}$$_{′}$ t − i k $^{′}$· x ( δ ( k − $_{k}$′$_{) +}$ 〈 ˆ a + k ′ ˆ a − k 〉) + e iω$_{k}$t − i k · x − iω$_{k}$$_{′}$ t + i k $^{′}$· x 〈 ˆ a + $_{k}$ˆ $_{a}$− k ′ 〉) d $^{3}$k d 3 $_{k}$′ (A.7) 〈 φ 2 〉 = 1 8 π 3 ∫ 1 ω$_{k}$ ( 1 2 + n k ) d $^{3}$k (A.8) = 1 8 π 3 ∫ ∞ 0 k 2 ω$_{k}$ ( 1 2 + n k ) d k ∫ π 0 sin( θ ) d θ ∫ 2 π 0 d ϕ (A.9) = 1 2 π 2 ∫ ∞ 0 k 2 ω$_{k}$ ( 1 2 + n k ) d k (A.10) Working through the integral, we obtain where n k is the occupation number, as well as the commutation relation we obtain the 2-point correlator as