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The subject of this thesis is the coupling of quantum fields to a classical gravitational background in a semiclassical fashion.
It contains a thorough introduction into quantum field theory on curved spacetime with a focus on the stress-energy tensor and the semiclassical Einstein equation.
Basic notions of differential geometry, topology, functio...
Citations
... this perturbed metric, should give us a Poisson tensor of next order in the deformation parameter. This approach is analogous to solving the semi-classical Einstein equations, see [Hac10,Pin11,Sie15,JA21] and references therein. ...
We introduce a non-commutative product for curved spacetimes, that can be regarded as a generalization of the Rieffel (or Moyal-Weyl) product. This product employs the exponential map and a Poisson tensor, and the deformed product maintains associativity under the condition that the Poisson tensor Θ satisfies \Theta \Nabla \Theta=0, in relation to a Levi-Cevita connection. We proceed to solve the associativity condition for various physical spacetimes, uncovering non-commutative structures with compelling properties.
... Ongoing, the first result toward a mathematical solution theory, providing local existence and uniqueness results for the trace of the SCE, was formulated in the seminal article [42] by Pinamonti. This approach was further refined, particularly studying global properties of solutions and their continuability, by Pinamonti and Siemssen in [43,46]. These works, similarly to many of the older references cited above, focused the conformally coupled case. ...
... (in conformal-time cosmological coordinates). For this particular representation of the Bunch-Davies state's two-point function, we refer to [46]; see also [2] for a similar representation. As a remark, we note that ν is not necessarily real. ...
... By this scheme, the QSE tensor indeed obeys ∇ μ T ren μν ω = 0. Finally, we add the renormalization freedom c 1 m 4 g μν + c 2 m 2 G μν + c 3 I μν + c 4 J μν in terms of four independent parameters c 1 , c 2 , c 3 , c 4 . We refer to [44,46] and references therein for precise formulas regarding H, ν 1 , I μν and J μν . Note that for an explicit expression for H one has to introduce a length scale, the so-called Hadamard length scale, in order to make the arguments of some occurring logarithmic dependencies unit free. ...
Exponentially expanding space–times play a central role in contemporary cosmology, most importantly in the theory of inflation and in the dark energy driven expansion in the late universe. In this work, we give a complete list of de Sitter solutions of the semiclassical Einstein equation (SCE), where classical gravity is coupled to the expected value of a renormalized stress–energy tensor of a free quantum field in the Bunch–Davies state. To achieve this, we explicitly determine the stress–energy tensor associated with the Bunch–Davies state using the recently proposed “moment approach” on the cosmological coordinate patch of de Sitter space. From the energy component of the SCE, we thus obtain an analytic consistency equation for the model’s parameters which has to be fulfilled by solutions to the SCE. Using this equation, we then investigate the number of solutions and the structure of the solution set in dependency on the coupling parameter of the quantum field to the scalar curvature and renormalization constants using analytic arguments in combination with numerical evidence. We also identify parameter sets where multiple expansion rates separated by several orders of magnitude are possible. Potentially for such parameter settings, a fast (semi-stable) expansion in the early universe could be compatible with a late-time “Dark Energy-like” behavior of the universe.
... The second goal of semiclassical gravity is to solve these equations and to nd the improved dynamics of the classical gravitational eld that consistently includes the backreaction of the quantum matter uctuations. The following summary of the basic ideas and results in semiclassical gravity is mainly based on the the textbooks and articles by Ford (2005), Hack (2016), Hu and Verdaguer (2020), and Siemssen (2015). Note also that parts of this summary can be found in (Schander and Thiemann 2021). ...
... For applications in cosmology, the so-called adiabatic regularization procedure (Fulling, Parker, and Hu 1974a,b;Parker and Fulling 1974) is another way to make sense of the formal expression (∶ ∶). This procedure is essentially equivalent to the above Hadamard pointsplitting regularization, in particular, they di er only by local curvature tensors (Siemssen 2015). It relies on the use of adiabatic states (Parker 1969) which are only approximately Hadamard but their straightforward construction proves to be useful (Junker and Schrohe 2002). ...
... Applications of the semiclassical scheme to cosmological situations are numerous, and we can only discuss a small fraction of them here. In general, these works aim at estimating the backreaction due to matter quantum elds on the cosmological background, mainly restricted to the conformally coupled scalar eld case (Siemssen 2015). One kind of e ects is related to the non-vanishing trace (hence denoted as trace anomaly) of the stress-energy tensor for quan-tum elds (Hu and Verdaguer 2020): In fact, for massless conformally coupled elds, the only quantum source in the semiclassical Einstein equations comes from the trace. ...
The present thesis addresses the problem of cosmological backreaction, i.e., the question of whether and to which extent cosmological inhomogeneities affect the global evolution of the Universe. We will thereby focus on, but not restrict to, backreaction in a purely quantum theoretical framework which is adapted to describe situations during the earliest phases of the Universe. Our approach to evaluating backreaction uses a perturbative and constructive mathematical formalism which is denoted as space adiabatic perturbation theory, and which extends the well-known Born-Oppenheimer approximation to molecular systems. The underlying idea of this scheme is to separate the system into an adiabatically slow and a fast part, similar to the separation of nuclear and electronic subsystems in a molecular setting. Such a distinction is reasonable if a corresponding perturbation parameter can be identified. In case of molecular systems, such a parameter arises as the ratio of the light electron and heavy nuclear masses. In the case of the here considered cosmological systems, we identify the ratio of the gravitational and the matter coupling constants as a suitable perturbative parameter. In a first step, we apply the space adiabatic formalism to a toy model and compute the backreaction of a homogeneous scalar field on a homogeneous and isotropic geometry. We restrict the computations to second order in the adiabatic perturbations and obtain an effective Hamilton operator for the geometry. In the sequel, we apply space adiabatic perturbation theory to an inhomogeneous cosmology and calculate backreaction effects of the inhomogeneous quantum cosmological fields on the global quantum degrees of freedom. Therefore, it is necessary to first extend the scheme adequately for an application to infinite dimensional field theories. In fact, the violation of the Hilbert-Schmidt condition for quantum field theories prevents a direct application of the scheme. A solution is obtained by a transformation of variables which is canonical up to second order in the cosmological perturbations. This allows us to compute an effective Hamilton operator for a cosmological field theory previously deparametrized by a timelike dust field as well as the identification of an effective Hamilton constraint for a system with gauge-invariant cosmological perturbations. Both objects act on the global degrees of freedom and include the backreaction of the inhomogeneities up to second order in the adiabatic perturbation theory. We conclude that it is a priori inadmissible to neglect cosmological backreaction. However, due to the general difficulties associated with finding solutions for coupled gravitational systems, the concrete evaluation of the operators found here must remain the subject of future research. One obstacle is the occurrence of indefinite mass squares associated with the perturbation fields which are the result of the previous transformations (which however, already appear in independent problems, for example in the use of Mukhanov-Sasaki variables) . A further complication in the final quantization and search for appropriate solutions arises from the non--polynomial dependence on the global degrees of freedom. We discuss these obstacles in detail and point to possible solutions.
... More precisely, in view of (17), the relevant observables that we need to control are the Wick square :φ 2 : and the energy density : :. Their expectation values can be obtained following the analyses performed in [18,19,26,56,58] and in the state (12) they take the form ...
... where X 0 = X(τ 0 ) and C is a suitable constant. The first step is to rewrite (58) in terms of the dynamic variable X , in order to obtain the explicit expression of the map C. ...
... Lemma 5.7. Given the initial data (a 0 , a 0 , X 0 , X 0 ), chosen in such a way that a 0 > 0 and Ω 2 k (τ 0 ) in (13) is strictly positive, and a state ω which is regular and compatible with this initial conditions, the semiclassical equation (58) can be written in the form of a fixed-point equation on C[τ 0 , τ 1 ] ...
We prove existence and uniqueness of solutions of the semiclassical Einstein equation in flat cosmological spacetimes driven by a quantum massive scalar field with arbitrary coupling to the scalar curvature. In the semiclassical approximation, the backreaction of matter to curvature is taken into account by equating the Einstein tensor to the expectation values of the stress-energy tensor in a suitable state. We impose initial conditions for the scale factor at finite time, and we show that a regular state for the quantum matter compatible with these initial conditions can be chosen. Contributions with derivative of the coefficient of the metric higher than the second are present in the expectation values of the stress-energy tensor and the term with the highest derivative appears in a non-local form. This fact forbids a direct analysis of the semiclassical equation, and in particular, standard recursive approaches to approximate the solution fail to converge. In this paper, we show that, after partial integration of the semiclassical Einstein equation in cosmology, the non-local highest derivative appears in the expectation values of the stress-energy tensor through the application of a linear unbounded operator which does not depend on the details of the chosen state. We prove that an inversion formula for this operator can be found, furthermore, the inverse happens to be more regular than the direct operator and it has the form of a retarded product, hence, causality is respected. The found inversion formula applied to the traced Einstein equation has thus the form of a fixed point equation. The proof of local existence and uniqueness of the solution of the semiclassical Einstein equation is then obtained applying the Banach fixed point theorem.
... That is, using Hadamard point-splitting, we have for the expectation value of the stress-energy tensor in a state ω (cf. [31,41,53]): ...
... uniquely defines Synge's world function. An expansion of Synge's world function σ(x, y) in terms of the coordinate distance δx between the points x and y can be obtained in the following way [53]: We make the Ansatz (in the sense of formal power series) σ(x, y) = n 1 n! ς μ1···μn (x)δx μ1 δx μn . ...
We develop a comprehensive framework in which the existence of solutions to the semiclassical Einstein equation (SCE) in cosmological spacetimes is shown. Different from previous work on this subject, we do not restrict to the conformally coupled scalar field and we admit the full renormalization freedom. Based on a regularization procedure, which utilizes homogeneous distributions and is equivalent to Hadamard point splitting, we obtain a reformulation of the evolution of the quantum state as an infinite-dimensional dynamical system with mathematical features that are distinct from the standard theory of infinite-dimensional dynamical systems (e.g., unbounded evolution operators). Nevertheless, applying methods closely related to Ovsyannikov’s method, we show existence of maximal/global solutions to the SCE for vacuum-like states and of local solutions for thermal-like states. Our equations do not show the instability of the Minkowski solution described by other authors.
... More precisely, in view of (17), the relevant observables that we need to control are the Wick square :φ 2 : and the energy density :̺:. Their expectation values can be obtained following the analyses performed in [17,18,25,52,54] and in the state (12) they take the form ...
... In this section we shall present the main result of this paper, namely the existence and uniqueness of solutions of the semiclassical Einstein equation (42) for a fixed arbitrary coupling parameter ξ = 1/6. We shall use all the results previously obtained in order to translate the original semiclassical equation in the form given in (53) into an of the form (54). We shall use the continuity property of the inverse operator T −1 0 given in (63) and proved in Proposition 5.3 in order to define a suitable contraction map. ...
We prove existence and uniqueness of solutions of the semiclassical Einstein equation in flat cosmological spacetimes driven by a quantum massive scalar field with arbitrary coupling to the scalar curvature. In the semiclassical approximation, the backreaction of matter to curvature is taken into account by equating the Einstein tensor to the expectation values of the stress-energy tensor in a suitable state. We impose initial conditions for the scale factor at finite time and we show that a regular state for the quantum matter compatible with these initial conditions can be chosen. Contributions with derivative of the coefficient of the metric higher than the second are present in the expectation values of the stress-energy tensor and the term with the highest derivative appears in a non-local form. This fact forbids a direct analysis of the semiclassical equation, and in particular, standard recursive approaches to approximate the solution fail to converge. In this paper we show that, after partial integration of the semiclassical Einstein equation in cosmology, the non-local highest derivative appears in the expectation values of the stress-energy tensor through the application of a linear unbounded operator which does not depend on the details of the chosen state. We prove that an inversion formula for this operator can be found, furthermore, the inverse happens to be more regular than the direct operator and it has the form of a retarded product, hence causality is respected. The found inversion formula applied to the traced Einstein equation has thus the form of a fixed point equation. The proof of local existence and uniqueness of the solution of the semiclassical Einstein equation is then obtained applying the Banach fixed point theorem.
... That is, using Hadamard point-splitting, we have for the expectation value of the stress-energy tensor in a state ω (cf. [25,31,42]): ...
... [σ] = 0, [(∇ µ ⊗ 1)σ] = 0, [(∇ µ ∇ ν ⊗ 1)σ] = g µν (A.11) uniquely defines Synge's world function. An expansion of Synge's world function σ(x, x ′ ) in terms of the coordinate distance δx between the points x and x ′ can be obtained in the following way [42]: We make the Ansatz (in the sense of formal power series) σ(x, x ′ ) = n 1 n! ς µ 1 ···µ n (x)δx µ 1 δx µ n . ...
We develop a comprehensive framework in which the existence of solutions to the semiclassical Einstein equation (SCE) in cosmological spacetimes is shown. Different from previous work on this subject, we do not restrict to the conformally coupled scalar field and we admit the full renormalization freedom. Based on a regularization procedure, which utilizes homogeneous distributions and is equivalent to Hadamard point-splitting, we obtain a reformulation of the evolution of the quantum state as an infinite-dimensional dynamical system with mathematical features that are distinct from the standard theory of infinite-dimensional dynamical systems (e.g. unbounded evolution operators). Nevertheless, applying new mathematical methods, we show existence of maximal/global (in time) solutions to the SCE for vacuum-like states, and of local solutions for thermal-like states. Our equations do not show the instability of the Minkowski solution described by other authors.
In this chapter we discuss two cosmological applications of algebraic quantum field theory in curved spacetimes. In the Standard Model of Cosmology—the CDM-model—the matter-energy content of the universe on large scales is modelled by a classical stress-energy tensor of perfect fluid form. Motivated by the fact that this matter-energy is considered to have a microscopic description in terms of a quantum field theory, we demonstrate as a first application how the classical perfect fluid stress-energy tensor in the CDM-model may be derived within quantum field theory on curved spacetimes by showing that there exist quantum states on cosmological spacetimes in which the expectation value of the quantum stress-energy tensor is qualitatively and quantitatively of the form assumed in the CDM-model up to corrections which may have interesting phenomenological implications. In the simplest models of Inflation, it is assumed that a classical scalar field on a cosmological spacetime coupled to the metric via the Einstein equations drives an exponential phase of expansion in the early universe. As a second application, the standard approach to the quantization of the perturbations of this coupled system, which makes heavy use of the symmetries of cosmological spacetimes, is re-examined by comparing it with a more fundamental approach which consists of quantizing the perturbations of a scalar field and the metric field in a gauge-invariant manner on general backgrounds and then considering the symmetric cosmological backgrounds as a special case.
This monograph provides a largely self--contained and broadly accessible
exposition of two cosmological applications of algebraic quantum field theory
(QFT) in curved spacetime: a fundamental analysis of the cosmological evolution
according to the Standard Model of Cosmology and a fundamental study of the
perturbations in Inflation. The two central sections of the book dealing with
these applications are preceded by sections containing a pedagogical
introduction to the subject as well as introductory material on the
construction of linear QFTs on general curved spacetimes with and without gauge
symmetry in the algebraic approach, physically meaningful quantum states on
general curved spacetimes, and the backreaction of quantum fields in curved
spacetimes via the semiclassical Einstein equation. The target reader should
have a basic understanding of General Relativity and QFT on Minkowski
spacetime, but does not need to have a background in QFT on curved spacetimes
or the algebraic approach to QFT. In particular, I took a great deal of care to
provide a thorough motivation for all concepts of algebraic QFT touched upon in
this monograph, as they partly may seem rather abstract at first glance. Thus,
it is my hope that this work can help non--experts to make `first contact' with
the algebraic approach to QFT.
