Mode decomposition and renormalization in semiclassical gravity

We compute the influence action for a system perturbatively coupled to a linear scalar field acting as the environment. Subtleties related to divergences that appear when summing over all the modes are made explicit and clarified. Being closely connected with models used in the literature, we show how to completely reconcile the results obtained in the context of stochastic semiclassical gravity when using mode decomposition with those obtained by other standard functional techniques.

We compute the influence action for a system perturbatively coupled to a linear scalar field acting as the environment. Subtleties related to divergences that appear when summing over all the modes are made explicit and clarified. Being closely connected with models used in the literature, we show how to completely reconcile the results obtained in the context of stochastic semiclassical gravity when using mode decomposition with those obtained by other standard functional techniques. 04.62.+v, 05.40.-a, 98.80.Cq

I. INTRODUCTION
The Closed Time Path (CTP) functional formalism has been very useful to study the back-reaction effect in the context of semiclassical gravity (as well as other aspects of field theory) [1][2][3][4]. When considering the back-reaction problem in semiclassical gravity, one is usually interested only in the gravitational field dynamics whereas the quantum matter fields are treated as an environment [6,7]. The results obtained when integrating out the environment degrees of freedom are closely connected with the influence functional [5], a statistical field theory method which has proved very fruitful to reveal the stochastic nature of open quantum systems (for applications to quantum Brownian motion models see Ref. [8]). In fact, it has been pointed out that semiclassical gravity [9,6] and effective theories in general should exhibit dissipation and noise [4]. To describe the stochastic character of the system dynamics due to the noise induced by the environment, Langevin-type equations are required. Thus, Einstein-Langevin equations have been used to address the back-reaction problem in the framework of semiclassical gravity [6,10,7,11].
When dealing with fields, mode decomposition can be a useful calculational tool since it makes the problem closer to quantum mechanical systems (free fields are treated as an infinite set of decoupled harmonic oscillators). The main advantage of this method is that the noise and dissipation kernels can be obtained in a rather direct way [12] and, in the context of semiclassical gravity, provides a simple connection with the Bogoliubov coefficients (closely related to particle creation effects) [6,10]. For each mode no renormalization is required, the need for renormalization arises when considering an infinite number of degrees of freedom: it is precisely when summing over all the modes that one gets ultraviolet divergences. However, the appearance of distributional functions makes this sum rather subtle, the presence of such divergences is not always manifestly evident and misleading results may be obtained. In the semiclassical gravity context this is particularly important as one may overlook the need for counterterms to renormalize the divergences, which will imply the appearance of finite extra terms when addressing the back-reaction problem. These drawbacks do not arise in other treatments based on functional methods typical of quantum field theory which make no use of mode decomposition, where renormalization seems to be more easily handled [1,7,13].
The aim of this report is to show how to reconcile the results obtained by means of a mode-decomposition approach with the results based on standard field theory techniques for renormalization in curved space-times [13]. In Sect. II we introduce the notation and the model that we are going to work with and evaluate the influence action perturbatively. A concrete example is considered in Sect. III, where sum over modes is performed revealing the appearance of divergences, and it is shown how they can be handled. In Sect. IV, the previous results are used to consider models treated in the literature which use mode decomposition in the context of stochastic semiclassical gravity and show how to reconcile these results with those obtained by usual functional methods.

II. MODE-DECOMPOSED EXPRESSION FOR THE INFLUENCE ACTION
To make the description as simple as possible we follow ref. [6] and consider the whole action for a system described by the variable x(t) with action S[x(t)] and the environment, described by a free field φ(t, x) in flat space which has been decomposed in a complete set of modes {u k ( x)}: φ(t, x) = k q k (t)u k ( x). The free action S[φ(t, x)] is local and at most quadratic in φ(t, x), since the field is linear. If that is also the case for the term S int [x(t), φ(t, x)] describing the interaction with the system, the action terms for the environment may be written, after performing the spatial integrals and using the completeness relation for the modes, as k S[q k (t)] and k S int k [x(t), q k (t)] respectively, where the action for each mode S[q k (t)] corresponds to that of a harmonic oscillator. The dynamics is therefore equivalent to that of a set of decoupled harmonic oscillators interacting separately with the system.
with Q k and h being some specific functions. The expression for the Feynman and Vernon influence functional [5] in the interaction picture is: where T + and T − correspond to the time ordering and anti-time ordering prescriptions respectively. To obtain the influence action S IF , we will treat the interaction term S int k [x(t)] perturbatively. Taking the logarithm of (2) and expanding up to second order in S int k , we get: where ) . All the expectation values are considered with respect to the asymptotic in vacuum |0 in I in the interaction picture. Note that we have integrated out the environment degrees of freedom and S IF depends only on the system variables.
It is important to separate the real and imaginary parts of the influence action because, as is well known [5,6], the imaginary part is related to the noise that the environment induces on the system, whereas the real part gives the averaged dynamics of the system. These are: where we have defined Σ (t) ≡ h (x + (t)) + h (x − (t)) and ∆ (t) ≡ h (x + (t)) − h (x − (t)) and we have introduced which has been expressed in two alternative and equivalent ways for further use. Here the kernels A k , D k and N k are defined as follows: ) is the noise kernel. The dissipation and noise kernels, which are related by the fluctuation-dissipation theorem, are antisymmetric and symmetric respectively under interchange of t and t ′ . On the other hand, the kernel A k (t, ] is symmetric and, as we will see, it is the part that gives rise to divergences.

III. SUM OVER ALL THE MODES AND NEED FOR RENORMALIZATION
For concreteness, let us now consider the case Q k (q k (t)) = g 2 q k (t) 2 (g is a perturbative coupling constant) where φ(t, x) is a massless real scalar field satisfying the Klein-Gordon equation in Minkowski space-time. We can use the following conventions (note that the label for each mode, k, corresponds in fact to a three dimensional vector): , whereâ † k andâ k are the creation and annihilation operators for each of the modes u k ( x) in which the field φ(t, x) has been decomposed. When properly normalized, f k (t) = (2π) − 3 2 (2ω k ) − 1 2 exp(−iω k t) with ω k = ( k 2 ) 1 2 . Taking all this into account, we will have To perform the sum over all the modes, we note that we may write Using the integral representations for When carrying out the sum over modes ( k ≡ V /(2π) n−1 d n−1 k ) we note that the G 2 F term will diverge. Thus, we use dimensional regularization to perform the integrals and then expand in powers of (n − 4), where n is the spacetime dimension. The usual procedure gives: where A = k A k , D = k D k and N = k N k . The second integral is finite and thus, we finally have the finite parts: 16π(n − 4))δ(t − t ′ ) has been separated in such a way that the divergences may be absorbed by counterterms in S[x]. In other QFT contexts (e.g. two interacting scalar fields) [2,4] the finite contribution from the counterterms can be reabsorbed in the renormalized parameters. However, as we will see, in semiclassical gravity some logarithmic finite terms which cannot be reabsorbed arise in the counterterms.

IV. STOCHASTIC SEMICLASSICAL GRAVITY
As an example we consider the back reaction due to the effect of a small mass or a non-conformal coupling of the scalar field φ(x) on a flat Robertson-Walker model [6,10,13]. We have to make the following substitutions: where η is the conformal time, a(η) the scale factor, R(a(η)) the scalar curvature, m the scalar field mass and ξ a dimensionless constant. In those previous works where Einstein-Langevin equations were derived using mode decomposition, divergences were not dealt with [6,10].
Let us now see how special care is needed with the sum of modes. Take for instance the second definition for H k in (6) and note that, using the real part of (13), D(η − η ′ ) = k D k (η − η ′ ) may be written as (V /16π 2 ) PV(1/(η − η ′ )). In this case, one would be inclined to write H = k H k = D(η, η ′ ) · θ(η − η ′ ) = (V /16π 2 ) PV(1/(η − η ′ )) · θ(η − η ′ ), but this is an ill-defined product of distributions which may give rise to divergences. A possible way to deal with this is by using, instead, the first definition in (6) and consider A and D separately: where the first term in the last member will be ultraviolet divergent whereas the last term is finite. Now the divergence can be clearly identified and one may use the proper counterterm in dimensional regularization to cancel it: The second term in this integral, which is finite, will cause the appearance of extra terms when deriving the Einstein-Langevin equation. Using now the results of the previous section, we get total agreement with those results reached by functional methods which do not use mode separation [13]. A very interesting connection between dissipation and fluctuations in the metric and particle creation has been revealed by Calzetta and Hu [6]. They computed the energy dissipated by the gravitational field per unit volume as ρ d = dηdη ′ (∂H(η, η ′ )/∂η)∆ω 2 (η)∆ω 2 (η ′ ) (for simplicity we have considered that the asymptotic values of the scale factor are a in = a out = 1), and showed that it was equal to the energy density of the created particles, ρ created particles = (2π) −3 4πν 2 V ν|β ν | 2 dν , where β ν is the Bogoliubov coefficient for the modes with frequency ν. However, formal use of divergent expressions was made in such a derivation. Our treatment shows clearly that the divergent part A(η, η ′ ) of the kernel H(η, η ′ ) decomposed according to (15) gives no contribution since it is symmetric under interchange of η and η ′ and hence the derivative will be antisymmetric: This integral is, therefore, manifestly finite and can be computed using the dissipation kernel obtained from eq. (13) thus leading to the same result of Ref. [6] without need to deal with divergent expressions.