We discuss the renormalisation of the initial value problem in quantum field theory using the two-particle irreducible (2PI) effective action formalism. The nonequilibrium dynamics is renormalised by counterterms determined in equilibrium. We emphasize the importance of the appropriate choice of initial conditions and go beyond the Gaussian initial density operator by defining self-consistent initial conditions. We study the corresponding time evolution and present a numerical example which supports the existence of a continuum limit for this type of initial conditions.
We discuss the renormalisation of mixed 3-point functions involving tensorial and scalar operators in conformal field theories of general dimension. In previous work we analysed correlators of either purely scalar or purely tensorial operators, in each case finding new features and new complications: for scalar correlators, renormalisation leads to beta functions, novel conformal anomalies of type B, and unexpected analytic structure in momentum space; for correlators of stress tensors and/or conserved currents, beta functions vanish but anomalies of both type B and type A (associated with a $…
We compute the renormalised energy momentum tensor of a free scalar field coupled to gravity on an (n+1)-dimensional Einstein Static Universe (ESU), RxS^n, with arbitrary low energy effective operators (up to mass dimension n+1). A generic class of regulators is used, together with the Abel-Plana formula, leading to a manifestly regulator independent result. The general structure of the divergences is analysed to show that all the gravitational couplings (not just the cosmological constant) are renormalised for an arbitrary regulator. Various commonly used methods (damping function, point-spli…
From the Wilsonian point of view, renormalisable theories are understood as submanifolds in theory space emanating from a particular fixed point under renormalisation group evolution. We show how this picture precisely applies to their gravity duals. We investigate the Hamilton-Jacobi equation satisfied by the Wilson action and find the corresponding fixed points and their eigendeformations, which have a diagonal evolution close to the fixed points. The relevant eigendeformations are used to construct renormalised theories. We explore the relation of this formalism with holographic renormalisa…
Generalising a result of classical mechanics an infinite set of conserved quantities can be found for the bare equations of motion describing the evolution of a scalar field in out of equilibrium quantum field theory, in the large N approximation, with initial conditions corresponding to a thermal system of the free Hamiltonian. Using these new conserved quantities, sum-rules relating integrals over the mode-functions (momenta) can be derived. More, the corresponding renormalised quantities can also be computed out thus giving information about the evolution of the already known renormalised e…
We study the approach to equilibrium for a scalar field which is coupled to a large thermal bath. Our analysis of the initial value problem is based on Kadanoff-Baym equations which are shown to be equivalent to a stochastic Langevin equation. The interaction with the thermal bath generates a temperature-dependent spectral density, either through decay and inverse decay processes or via Landau damping. In equilibrium, energy density and pressure are determined by the Bose-Einstein distribution function evaluated at a complex quasi-particle pole. The time evolution of the statistical propagator…
We show that a non-zero renormalised value of the zero-point energy in $λϕ^4$-theory over Minkowski spacetime is in tension with the scalar-field equation at two-loop order in perturbation theory.
We review different computation methods for the renormalised energy momentum tensor of a quantised scalar field in an Einstein Static Universe. For the extensively studied conformally coupled case we check their equivalence; for different couplings we discuss violation of different energy conditions. In particular, there is a family of masses and couplings which violate the weak and strong energy conditions but do not lead to spacelike propagation. Amongst these cases is that of a minimally coupled massless scalar field with no potential. We also point out a particular coupling for which a mas…
We elaborate further the functional Schrödinger-picture approach to the quantum field in curved spacetimes using the generalized invariant method and construct explicitly the Fock space, which we relate with the thermal field theory. We apply the method to a free massive scalar field in the de Sitter spacetime, and find the exact quantum states, construct the Fock space, and evaluate the two-point function and correlation function.
Massless minimally coupled quantum scalar field with an asymmetric self interaction, $V(ϕ)=λϕ^4/4!+ βϕ^3/3! $ (with $λ>0$) is considered in the $(3+1)$-dimensional inflationary de Sitter spacetime. This potential is bounded from below irrespective of the sign of $β$. Earlier computations mostly considered the quartic part. Our chief motivation behind this study is to assess the vacuum expectation values of $V(ϕ)$ and $ϕ$, both of which can be negative, and hence may lead to some screening of the inflationary cosmological constant value. First using the in-in formalism, the renormalised quantum…
Real-time perturbation theory is formulated for complex scalar fields away from thermal equilibrium in such a way that dissipative effects arising from the absorptive parts of loop diagrams are approximately resummed into the unperturbed propagators. Low order calculations of physical quantities then involve quasiparticle occupation numbers which evolve with the changing state of the field system, in contrast to standard perturbation theory, where these occupation numbers are frozen at their initial values. The evolution equation of the occupation numbers can be cast approximately in the form …
We examine the precise connection between the exact renormalisation group with local couplings and the renormalisation of correlation functions of composite operators in scale-invariant theories. A geometric description of theory space allows us to select convenient non-linear parametrisations that serve different purposes. First, we identify normal parameters in which the renormalisation group flows take their simplest form; normal correlators are defined by functional differentiation with respect to these parameters. The renormalised correlation functions are given by the continuum limit of …
We consider the time evolution of systems in which a spatially homogeneous scalar field is coupled to fermions. The quantum back-reaction is taken into account in one-loop approximation. We set up the basic equations and their renormalization in a form suitable for numerical computations. The initial singularities appearing in the renormalized equations are removed by a Bogoliubov transformation. The equations are then generalized to those in a spatially flat Friedmann-Robertson-Walker universe. We have implemented the Minkowski space equations numerically and present results for the time evol…
Motivated by the initial value problem in semiclassical gravity, we study the initial value problem of a system consisting of a quantum scalar field weakly interacting with a classical one. The quantum field obeys a Klein-Gordon equation with a potential proportional to the classical field. The classical field obeys an inhomogeneous Klein-Gordon equation sourced by the renormalised expectation value of the squared quantum field in a Hadamard state, $\langle Ψ| Φ^2 Ψ\rangle$. Thus, the system of equations for the scalar fields reminisces of the semi-classical Einstein field equations with a Kle…
Radiatively induced SU(2) symmetry breaking is shown to be a genuine feature of SU(2) x O(N) globally symmetric renormalisable field theories in the large N limit, describing interaction of a complex SU(2) doublet, O(N)-singlet field with an SU(2) singlet, O(N) vector. Symmetry breaking solutions are found even when all fields have positive renormalised squared mass. The emerging novel mechanism of symmetry breaking can reproduce with a choice of N~300 the standard range of the electroweak condensate and the Higgs mass occurring in the extended Higgs dynamics of an SU(2) symmetric Gauge+Higgs …
We derive the renormalized nonequilibrium equations of motion for a scalar field and its quantum back reaction in a conformally flat Friedmann-Robertson-Walker universe. We use a fully covariant formalism proposed by us recently for handling numerically and analytically nonequilibrium dynamics in one-loop approximation. The system is assumed to be in a conformal vacuum state initially. We use dimensional regularization; we find that the counter terms can be chosen independent of the initial conditions though the divergent leading order graphs do depend on them.
We consider one-particle reducible (1PR) contributions to QED and scalar QED processes in external fields, at one-loop and two-loop order. We investigate three cases in detail: constant crossed fields, constant magnetic fields, and plane waves. We find that 1PR tadpole contributions in plane waves and constant crossed fields are non-zero, but contribute only divergences to be renormalised away. In constant magnetic fields, on the other hand, tadpole contributions give physical corrections to processes at one-loop and beyond. Our calculations are exact in the external fields and we give strong …
We examine the nonequilibrium dynamics of a self-interacting $λϕ^4$ scalar field theory. Using a real time formulation of finite temperature field theory we derive, up to two loops and $O(λ^2)$, the effective equation of motion describing the approach to equilibrium. We present a detailed analysis of the approximations used in order to obtain a Langevin-like equation of motion, in which the noise and dissipation terms associated with quantum fluctuations obey a fluctuation-dissipation relation. We show that, in general, the noise is colored (time-dependent) and multiplicative (couples nonlinea…
A partial resummation of perturbation theory is described for field theories containing spin-1/2 particles in states that may be far from thermal equilibrium. This allows the nonequilibrium state to be characterized in terms of quasiparticles that approximate its true elementary excitations. In particular, the quasiparticles have dispersion relations that differ from those of free particles, finite thermal widths and occupation numbers which, in contrast to those of standard perturbation theory evolve with the changing nonequilibrium environment. A description of this kind is essential for est…
We present a comprehensive discussion of renormalisation of 3-point functions of scalar operators in conformal field theories in general dimension. We have previously shown that conformal symmetry uniquely determines the momentum-space 3-point functions in terms of certain integrals involving a product of three Bessel functions (triple-K integrals). The triple-K integrals diverge when the dimensions of operators satisfy certain relations and we discuss how to obtain renormalised 3-point functions in all cases. There are three different types of divergences: ultralocal, semilocal and nonlocal, …
We derive the renormalized equations of motion and the renormalized energy-momentum tensor for fermions coupled to a spatially homogeneous scalar field (inflaton) in a flat FRW geometry. The fermion back reaction to the metric and to the inflaton field is formulated in one-loop approximation. Having determined the infinite counter terms in an $\bar{MS}$ scheme we formulate the finite terms in a form suitable for numerical computation. We comment on the trace anomaly which is inferred from the standard analysis. We also address the problem of initial singularities and determine the Bogoliubov t…
We present lattice simulations of nonequilibrium quantum fields in Minkowskian space-time. Starting from a non-thermal initial state, the real-time quantum ensemble in 3+1 dimensions is constructed by a stochastic process in an additional (5th) ``Langevin-time''. For the example of a self-interacting scalar field we show how to resolve apparent unstable Langevin dynamics, and compare our quantum results with those obtained in classical field theory. Such a direct simulation method is crucial for our understanding of collision experiments of heavy nuclei or other nonequilibrium phenomena in str…
We discuss the ringdown behavior of the nonequilibrium Green's function in a strongly coupled theory with the holographic dual with a focus on quasinormal-mode equilibration. We study the time resolved spectral function for a probe scalar in Vaidya-AdS spacetime in detail as a complement to the preceding work arXiv:1603.06935 using further numerical results in very nonadiabatic temperature changes. It is shown that the relaxation of the nonequilibrium spectral function obtained through the Wigner transform is governed by the lowest quasinormal mode frequency. The timescale of the background te…
In this paper, we will consider scattering diagrams in the context of infinite-derivative theories. First, we examine a finite-order higher-derivative scalar field theory and find that we cannot eliminate the external momentum divergences of scattering diagrams in the regime of large external momenta. Then, we employ an infinite-derivative scalar toy model and obtain that the external momentum dependence of scattering diagrams is convergent as the external momenta become very large. In order to eliminate the external momentum divergences, one has to dress the bare vertices of the scattering di…
Transport or kinetic equations are often derived assuming a quasi-particle (on-shell) representation of the spectral function. We investigate this assumption using a three-loop approximation of the 2PI effective action in real time, without a gradient expansion or on-shell approximation. For a scalar field in 1+1 dimensions the nonlinear evolution, including the integration over memory kernels, can be solved numerically. We find that a spectral function approximately described by a nonzero width emerges dynamically. During the nonequilibrium time evolution the Wigner transformed spectral funct…
We show that there exist solutions to the semi-classical gravity equations in de Sitter spacetime sourced by the renormalised stress-energy tensor of a free Klein-Gordon field. For the massless scalar, solutions exist for every possible value of the cosmological constant, provided that the curvature coupling parameter is chosen appropriately. In the massive case, imposing Wald's axioms for the renormalised stress-energy tensor, the mass of the field and the curvature coupling constraint the allowed values of $Λ$. For a massive, minimally coupled field, a "small $Λ$" solution is found, fixed by…
We quantify the quantum correlations between two accelerated detectors coupled to a scalar field in a cavity. It has been realized that an accelerated detector will experience a thermal bath, which is termed the Unruh effect. We examine the similarities and differences for quantum correlations regarding either temperature or acceleration. As the accelerations (resp. temperatures) of the detectors increase, the entanglement decreases to zero at some instant but the mutual information (resp. discord) can be amplified. As the accelerations increase but are in opposite directions, the quantum corr…
We study the dynamics of the holographic $s$-wave superconductors described by the Einstein-Maxwell-complex scalar field theory with a negative cosmological constant. If the eigenfunction of the linearized equation of motion of the scalar field in the planar RNAdS black hole background is chosen as the initial data, the bulk system will evolve to the intermediate state that corresponds to the excited state superconductor on the boundary. The process can be regarded as the non-equilibrium condensation process of the excited state of holographic superconductor. When the linear superposition of t…
We study the thermodynamics of $n$-dimensional static asymptotically AdS black holes in Einstein gravity coupled to a scalar field with a potential admitting a stationary point with an AdS vacuum. Such black holes with non-trivial scalar hair can exist provided that the mass-squared of the scalar field is negative, and above the Breitenlohner-Freedman bound. We use the Wald procedure to derive the first law of thermodynamics for these black holes, showing how the scalar hair (or "charge") contributes non-trivially in the expression. We show in general that a black hole mass can be deduced by i…
We present a general and model-independent method to obtain an effective Markovian quantum kinetic equation for the expectation value of a slowly evolving scalar field in an adiabatically evolving background from first principles of nonequilibrium quantum field theory. The method requires almost no assumptions about the field's interactions and the composition of the background, except that 1) the coupling constants shall be small enough for perturbation theory to be applicable, 2) there is a clear separation between microphysical time scales and the rate at which bulk properties change, and 3…
We have studied the decoherence mechanism in a fermion and scalar quantum field theory with the Yukawa interaction in the Minkowski spacetime, using the non-equilibrium effective field theory formalism appropriate for open systems. The scalar field is treated as the system whereas the fermions as the environment. As the simplest realistic scenario, we assume that an observer measures only the Gaussian 2-point correlator for the scalar field. The cause of decoherence and the subsequent entropy generation is the ignorance of information stored in higher-order correlators, Gaussian and non-Gaussi…
The flow equations of the renormalisation group permit to analyse the perturbative $n$-point functions of renormalisable quantum field theories. Rigorous bounds implying renormalisablility allow to control large momentum behaviour, infrared singularities and large order behaviour in the number of loops and the number of arguments $n\,$. Gauge symmetry which is broken by the flow in momentum or position space, can be shown to be restored in the renormalised theory. In this paper we want to do a first but important step towards a rigorous nonperturbative analysis of the flow equations (FEs). We …
We consider the non-equilibrium dynamics of a real quantum scalar field. We show the formal equivalence of the exact evolution equations for the statistical and spectral two-point functions with a fictitious Langevin process and examine the conditions under which a local Markovian dynamics is a valid approximation. In quantum field theory, the memory kernel and the noise correlator typically exhibit long time power laws and are thus highly non-local, thereby questioning the possibility of a local description. We show that despite this fact, there is a finite time range during which a local des…
We use the so-called Liouville-von Neumann (LvN) approach to study the nonequilibrium quantum dynamics of time-dependent second order phase transitions. The LvN approach is a canonical method that unifies the functional Schrödinger equation for the quantum evolution of pure states and the LvN equation for the quantum description of mixed states of either equilibrium or nonequilibrium. As nonequilibrium quantum mechanical systems we study a time-dependent harmonic and an anharmonic oscillator and find the exact Fock space and density operator for the harmonic oscillator and the nonperturbative …
The one-loop effective action for the scalar field part of a non-Abelian gauge theory based on a general gauge group of the form $G\times U(1)$, where the gauge group $G$ is arbitrary, is calculated. A complex scalar field, both Abelian and non-Abelian gauge fields and Dirac fermions coupled to gauge and scalar fields are included. A general mass term for the Dirac fields that includes a pseudoscalar term as well as both scalar and pseudoscalar Yukawa couplings is considered. The background field method is used in its manifestly gauge condition independent and gauge invariant form to isolate t…
We study the detailed out of equilibrium time evolution of a homogeneous Bose-Einstein condensate.We consider a nonrelativistic quantum theory for a self-interacting complex scalar field, immersed in a thermal bath, as an effective microscopic model for the description of the Bose-Einstein condensate. This approach yields the following main results:(i) the interaction between fluctuations proves to be crucial in the mechanism of instability generation; (ii) there are essentially two regimes in the $k$-space, with a crossover for $k^2/2m \sim 2λ|ϕ_0|^2$, where, in our notation, $λ$ is the coupl…
The nonequilibrium dynamics of quantum fields is an initial-value problem, which can be described by Kadanoff-Baym equations. Typically, and in particular when numerical solutions are demanded, these Kadanoff-Baym equations are restricted to Gaussian initial states. However, physical initial states are non-Gaussian correlated initial states. In particular, renormalizability requires the initial state to feature n-point correlations that asymptotically agree with the vacuum correlations at short distances. In order to identify physical nonequilibrium initial states, it is therefore a preconditi…
Density contrasts in the universe are governed by scalar cosmological perturbations which, when expressed in terms of gauge-invariant variables, contain a classical component from scalar metric perturbations and a quantum component from inflaton field fluctuations. It has long been known that the effect of cosmological expansion on a quantum field amounts to squeezing. Thus the entropy of cosmological perturbations can be studied by treating them in the framework of squeezed quantum systems. Entropy of a free quantum field is a seemingly simple yet subtle issue. In this paper, as different fro…
The hypothesis of quantum nonequilibrium at the big bang is shown to have observable consequences. For a scalar field on expanding space, we show that relaxation to quantum equilibrium (in de Broglie-Bohm theory) is suppressed for field modes whose quantum time evolution satisfies a certain inequality, resulting in a 'freezing' of early quantum nonequilibrium for these particular modes. For an early radiation-dominated expansion, the inequality implies a corresponding physical wavelength that is larger than the (instantaneous) Hubble radius. These results make it possible, for the first time, …
We consider $N$ uniformly-accelerating Unruh-DeWitt detectors whose internal degrees of freedom are coupled to a massless scalar field in $(1+1)$D Minkowski space. We use the influence functional formalism to derive the Langevin equations governing the nonequilibrium dynamics of the internal degrees of freedom and show explicitly that the system relaxes in time and equilibrates. We also show that once the equilibrium condition is established a set of fluctuation-dissipation relations (FDR) and correlation-propagation relations (CPR) emerges for the detectors, extending earlier results of [1] w…
Continuing our inquiry into the conditions when fluctuation-dissipation relations (FDR) may appear in the context of nonequilibrium dynamics of open quantum systems (over and beyond the conventional FDR from linear response theory) we turn to nonGaussian systems and consider this issue for an anharmonic oscillator interacting with a scalar quantum field bath. We present the general {nonperturbative} expressions for the rate of energy (power) exchange between the anharmonic oscillator and the thermal bath. For the cases that a stable final equilibrium state exists, and the nonstationary compone…
Nonperturbative dynamics of quantum fields out of equilibrium is often described by the time evolution of a hierarchy of correlation functions, using approximation methods such as Hartree, large N, and nPI-effective action techniques. These truncation schemes can be implemented equally well in a classical statistical system, where results can be tested by comparison with the complete nonlinear evolution obtained by numerical methods. For a 1+1 dimensional scalar field we find that the early-time behaviour is reproduced qualitatively by the Hartree dynamics. The inclusion of direct scattering i…
We consider a movable mirror coupled to a one-dimensional massless scalar field in a cavity. Both the field and the mirror's mechanical degrees of freedom are described quantum-mechanically, and they can interact each other via the radiation pressure operator. We investigate the dynamical evolution of mirror and field starting from a nonequilibrium initial state, and their local interaction which brings the system to a stationary configuration for long times. This allows us to study the time-dependent dressing process of the movable mirror interacting with the field, and its dynamics leading t…
Using a one-loop renormalization group improvement for the effective potential in the Higgs model of electrodynamics with electrically and magnetically charged scalar fields, we argue for the existence of a triple (critical) point in the phase diagram ($λ_{run}, g_{run}^4$), where $λ_{run}$ is the renormalised running selfinteraction constant of the Higgs scalar monopoles and $g_{run}$ is their running magnetic charge. This triple point is a boundary point of three first-order phase transitions in the dual sector of the Higgs scalar electrodynamics: The "Coulomb" and two confinement phases mee…
We discuss the renormalization of the initial value problem in Nonequilibrium Quantum Field Theory within a simple, yet instructive, example and show how to obtain a renormalized time evolution for the two-point functions of a scalar field and its conjugate momentum at all times. The scheme we propose is applicable to systems that are initially far from equilibrium and compatible with non-secular approximation schemes which capture thermalization. It is based on Kadanoff-Baym equations for non-Gaussian initial states, complemented by usual vacuum counterterms. We explicitly demonstrate how var…
We show that inflationary cosmology may be used to test the statistical predictions of quantum theory at very short distances and at very early times. Hidden-variables theories, such as the pilot-wave theory of de Broglie and Bohm, allow the existence of vacuum states with non-standard field fluctuations ('quantum nonequilibrium'). We show that inflationary expansion can transfer microscopic nonequilibrium to macroscopic scales, resulting in anomalous power spectra for the cosmic microwave background. The conclusions depend only weakly on the details of the de Broglie-Bohm dynamics. We discuss…
We consider a model of quantum-wire junctions where the latter are described by conformal-invariant boundary conditions of the simplest type in the multicomponent compactified massless scalar free field theory representing the bosonized Luttinger liquids in the bulk of wires. The boundary conditions result in the scattering of charges across the junction with nontrivial reflection and transmission amplitudes. The equilibrium state of such a system, corresponding to inverse temperature $β$ and electric potential $V$, is explicitly constructed both for finite and for semi-infinite wires. In the …
The quantum fluctuations of a test scalar field on superhorizon scale in de Sitter spacetime can be described by an effective one-dimensional stochastic theory corresponding to a particular class of nonequilibrium dynamical systems known as the model A. Using the formulation of the latter in terms of a supersymmetric field theory, we compute various unequal-time correlators at large (superhorizon) time separations and compare with existing quantum field theory computation. This includes perturbative calculations, pushed here up to three-loop order, and a nonperturbative $1/N$ expansion at next…
We propose a new approach to compute correlators of quantum fields in de Sitter space. It is based on nonequilibrium field theory techniques, and exploits de Sitter symmetries so as to partially reduce the number of independent variables of n-point functions in a manner that preserves the usefulness of a momentum representation, e.g., for writing spatial convolution integrals as simple products. In this representation, the two-point function of a scalar field only depends on two physical momenta, and the corresponding Schwinger-Dyson evolution equations take the form of momentum flow equations…
The possibility of maintaining entanglement in a quantum system at finite, even high, temperatures -- the so-called `hot entanglement' -- has obvious practical interest, but also requires closer theoretical scrutiny. Since quantum entanglement in a system evolves in time and is continuously subjected to environmental degradation, a nonequilibrium description by way of open quantum systems is called for. To identify the key issues and the contributing factors that may permit `hot entanglement' to exist, or the lack thereof, we carry out a model study of two spatially-separated, coupled oscillat…
We generalise results of Ford and Roman which place lower bounds -- known as quantum inequalities -- on the renormalised energy density of a quantum field averaged against a choice of sampling function. Ford and Roman derived their results for a specific non-compactly supported sampling function; here we use a different argument to obtain quantum inequalities for a class of smooth, even and non-negative sampling functions which are either compactly supported or decay rapidly at infinity. Our results hold in $d$-dimensional Minkowski space ($d\ge 2$) for the free real scalar field of mass $m\ge…
We solve the nonequilibrium dynamics of a 3+1 dimensional theory with Dirac fermions coupled to scalars via a chirally invariant Yukawa interaction. The results are obtained from a systematic coupling expansion of the 2PI effective action to lowest non-trivial order, which includes scattering as well as memory and off-shell effects. The dynamics is solved numerically without further approximation, for different far-from-equilibrium initial conditions. The late-time behavior is demonstrated to be insensitive to the details of the initial conditions and to be uniquely determined by the initial e…
We study incomplete relaxation to quantum equilibrium at long wavelengths, during a pre-inflationary phase, as a possible explanation for the reported large-scale anomalies in the cosmic microwave background (CMB). Our scenario makes use of the de Broglie-Bohm pilot-wave formulation of quantum theory, in which the Born probability rule has a dynamical origin. The large-scale power deficit could arise from incomplete relaxation for the amplitudes of the primordial perturbations. We show, by numerical simulations for a spectator scalar field, that if the pre-inflationary era is radiation dominat…
We calculate the effective potentials for scalar, Dirac and Yang-Mills fields in curved backgrounds using a new method for the determination of the heat kernel involving a re-summation of the Schwinger-DeWitt series. Self-interactions are treated both to one loop order as usual and slightly beyond one-loop order by means of a mean-field approximation. The new approach gives the familiar result for scalar fields, the Coleman-Weinberg potential plus corrections such as the leading-log terms, but the actual calculation is much faster. We furthermore show how to go systematically to higher loop or…
This is the second of a series of three papers examining how viable it is for entanglement to be sustained at high temperatures for quantum systems in thermal equilibrium (Case A), in nonequilibrium (Case B) and in nonequilibrium steady state conditions (Case C). The system we analyze here consists of two coupled quantum harmonic oscillators each interacting with its own bath described by a scalar field, set at temperatures $T_1 > T_2$. For \textit{constant bilinear inter-oscillator coupling} studied here (Case C1) owing to the Gaussian nature, the problem can be solved exactly at arbitrary te…
