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Chapter 6 Quantum Inflationary Perturbations

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Chapter 6 Quantum Inflationary Perturbations CHAPTER 6 QUANTUM INFLATIONARY PERTURBATIONS The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction. Sidney Coleman The procedure till now has been completely classical. In this Chapter we present the standard tools for the quantization of a scalar field in a non-trivial background. We will first apply the general treatment to the limiting case of a scalar field living in a rigid de Sitter spacetime. Finally we will quantize the curvature and tensor inflationary perturbations computed in the previous chapter. 6.1 Canonical quantization Consider the action for a scalar field with time-dependent mass Z Z 3 1 3 02 2 2 2 S = dτ d x = dτ d x v (@iv) m (τ)v ; (6.1) L 2 − − in some coordinates (τ; x). The first step in the canonical quantization procedure is to define the canonically conjugated momentum to the field v @ π = L = v0 ; (6.2) @v0 and to construct the Hamiltonian Z Z Z 3 3 0 1 3 2 2 2 2 H = d x = d x πv = d x π + (@iv) + m (τ)v : (6.3) H − L 2 In the quantum version of the theory the (classical) variables v and π become (quantum) operators satisfiying the equal-time conmutation relations [^v (x; τ) ; π^ (y; τ)] = i δ(x y) ; [^v (x; τ) ; v^ (y; τ)] = [^π (x; τ) ; π^ (y; τ)] = 0 ; (6.4) − 6.1 Canonical quantization 64 and H(v; π) H^ (^v; π^). The operatorv ^ obeys the same equation of motion as the classical ! variable v, namely v^00 2v^ + m2(τ)v ^ = 0 : (6.5) − r The general solution of this partial differential equation can be written as an infinite super- position of Fourier modes Z 3 d k ik·x v^ (x; τ) = vke (2π)3=2 Z 3 d k − + ∗ ik·x = a^ vk(τ) +a ^ v (τ) e (6.6) (2π)3=2 k −k k Z 3 d k − ik·x + ∗ −ik·x = a^ vk(τ)e +a ^ v (τ)e : (2π)3=2 k k k The temporal mode functions vk(τ) is this expression satisfy harmonic oscillator equations with time-dependent frequency 00 2 2 2 2 vk + !k(τ)vk = 0 ;!k(τ) = k + m (τ) : (6.7) Note that the mode functions vk(τ) depends only of k = k due to the isotropy of the FLRW 2 j 2j Universe and the associated form of the frequency !k k . ± ∼ The operatorsa ^k in Eq. (6.6) are taken to be time-independent and to obey the bosonic commutation relations − + 0 − − + + a^ ; a^ 0 = δ(k k ); a^ ; a^ 0 = a^ ; a^ 0 = 0 : (6.8) k k − k k k k The consistency of these commutation relations with those in (6.4) requires the mode func- tions vk(τ) to satisfy the normalization condition 0 ∗ ∗ 0 0 ∗ v v vkv = 2 i Im v v = i : (6.9) k k − k k k − Exercise Derive Eq. (6.9). Note that left hand side of this equation is nothing else than the ∗ 0 ∗ ∗ 0 Wronskian of the ordinary differential equation (6.7), W [vk; v ] v v vkv . k ≡ k k − k 6.1.1 Bogolyubov transformations ± The creation and annihilation operatorsa ^k can be used to construct an orthonormal basis in the Hilbert space m n − 1 h + + i a^ 0 a = 0 ; mk1 ; nk2 ; ::: a^ a^ ::: 0 a ; (6.10) k j i j ia ≡ pm!n!::: k1 k2 j i with 0 a the a-vacuum state for all k and mk ; nk ; ::: denoting a set of excited states with j i j 1 2 ia occupation numbers m; n; : : : in the modes k1; k2;::: Note, however, that an unambiguous interpretation of these states arises only after selecting the precise mode functions vk(τ) in 6.1 Canonical quantization 65 Eq. (6.6). Each choice of mode functions translates into a different set of creation and an- nihilation operators and consequently into different vacuum and excited states. To illustrate this, consider the linear combination ∗ uk(τ) αkvk(τ) + βkv (τ) ; (6.11) ≡ k with αk and βk time-independent complex coefficients. Requiring the modes uk to satisfy the normalization condition 0 ∗ ∗ 0 0 ∗ u u uku = 2 i Im u u = i : (6.12) k k − k k k − we otain the following relation between the complex coefficients αk and βk 2 2 αk βk = 1: (6.13) j j − j j Exercise Show this. The uk(τ) modes satisfy, by construction, the wave equation (6.7). This allows us to use them as an alternative basis to expand the fieldv ^, namely Z 3 d k − ik·x + ∗ −ik·x v^ (x; τ) = ^b uk(τ)e + ^b u (τ)e ; (6.14) (2π)3=2 k k k ^± with bk a different set of time-independent creation and annihilation operators satisfying the commutation relations h − + i 0 h − − i h + + i ^b ; ^b 0 = δ(k k ); ^b ; ^b 0 = ^b ; ^b 0 = 0 : (6.15) k k − k k k k ^± As before, the creation and annihilation operators bk can be used to construct and orthonor- mal basis in the Hilbert space m n ^− 1 h^+ ^+ i b 0 b = 0 ; mk1 ; nk2 ; ::: b b ::: 0 b : (6.16) k j i j ib ≡ pm!n!::: k1 k2 j i with 0 b the b-vacuum for all k and mk ; nk ; ::: a multiparticle state of b-particles. j i j 1 2 ia ± ^± The relation between the operatorsa ^k amd bk can be determined by substituting the linear combination (6.11) into the mode expansion (6.14) and comparing the result with Eq. (6.6). Doing this we obtain − ∗ ^− ^+ + ^+ ∗ ^− a^k = αk bk + βk b−k ; a^k = αk bk + βk b−k : (6.17) The expressions in (6.17) are called Bogolyubov transformations. The complex coefficients αk and βk are Bogolyubov coefficients. Exercise Derive (6.17). 6.1 Canonical quantization 66 The relation (6.17) can be used to compute the expectation number of the a-particle number operator N^ a a^+a^− in the b-vacuum k ≡ k k a + ∗ − ∗ − + 2 b 0 N^ 0 b = b 0 (αk ^b + β ^b )(α ^b + βk ^b ) 0 = βk δ(0) : (6.18) h j kj i h j k k −k k k −k j ib j j The factor δ(0) is just the infinite spatial volume. It arises from the commutation relations when evaluating δ(k k0) at k = k0. Dividing by δ(0) we obtain the number density of − particles in a given mode, 2 nk βk : (6.19) ≡ j j This expression shows that the b-vacuum is generically a state without b-particles but with an a-particle number density nk in each mode. Natural modes and Killing vectors In Minkowski spacetime there exist a natural set of modes which are closely related to the Poincar´egroup. Any other mode function mixing positive and negative frequency natural modes is not Poincar´einvariant. Bogolubov transformations that do not mix positive and negative energy modes are called trivial or non-mixing. The existence of natural modes in curved spacetimes is associated to the presence of time-like Killing vectors. 6.1.2 Choice of physical vacuum In the previous section we have seen that each vacuum state has a non-zero number density of particles for any other particle. Which is then the \true" physical vacuum? The usual Minkowski spacetime definition of the vacuum as the \lowest energy eigen- state" cannot be applied to the problem at hand. The Hamiltonian (6.3) involves an explicit function of time, meaning that the concept of \lowest energy eigenstate" depends on the particular instant τ0 at which it is defined. However, it is always possible to define an instan- taneous vacuum state at a given time τ0. To see this consider the expectation value of the Hamiltonian for an arbitrary mode function vk(τ). Inserting the mode expansion (6.6) into the quantized version of Eq. (6.3), we get Z ^ 3 ∗ − − + + + − H = d k Fk a^k a^−k + Fk a^k a^−k + Ek 2^ak a^k + δ(0) ; (6.20) with 1 0 2 2 2 1 02 2 2 Ek v + ! vk ;Fk v + ! v : (6.21) ≡ 2 j kj kj j ≡ 2 k k k Taking the expectation value of Eq. (6.20) removes the first three terms since each of them annihilates the vacuum on either the left- of the right-hand side. We are left with Z 3 v 0 H^ 0 v = δ(0) d k Ek : (6.22) h j j i Dividing by the δ(0)-term accounting for the infinite volume of space, we obtain the energy density Z 3 " d k Ek: (6.23) ≡ 6.1 Canonical quantization 67 The minimum of this quantity at a given time τ0 can be obtained by minimizing the individual contributions Ek at that time. To this end, let us consider the polar representation iαk vk = rke ; (6.24) with rk and αk real functions. Replacing this expression into Eq. (6.21) we get 1 02 2 02 2 2 1 02 1 2 2 Ek(τ0) = rk + rkαk + !krk = rk + 2 + wkrk ; (6.25) 2 2 4rk where in the last step we have made use of the Wronskian condition (6.9) in its polar form 0 ∗ ∗ 0 0 1 vkvk vkvk = i ; = αk = 2 : (6.26) − − ) −2rk Equation (6.25) is minimized for 0 1 rk(τ0) = 0 ; rk(τ0) = ; (6.27) p2!k which corresponds a choice of instantaneous mode functions 1 −i!k(τ0)τ0 0 vk (τ0) = p e ; vk(τ0) = i!kvk(τ0) : (6.28) 2!k(τ0) − ± These modes functions define a particular set of operators ak and the associated basis of vacuum and excited states.
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