Correspondences of matter field fluctuations in semiclassical and classical gravity in the decoherence limit

Correspondence between the stress energy tensor for scalar fields and that for the general fluids in the hydrostatic/hydrodynamic approximation is well known [1, 2]. The area of field fluid correspondence is open for investigations and there are ongoing efforts over various aspects [3, 4]. In this article, as a sequel of recent work [5] we show correspondence between the fluctuations of non-minimally coupled quantum fields and the effective general fluid in the decoherence limit. The result is obtained in terms of two point noise kernels, which can be worked out from the first principles using the respective stress tensors. We use the exact form of the semiclassical noise kernel [6, 7] defining fluctuations of non-minimally coupled massive quantum fields and relate these to the fluctuations of an effective fluid stress tensor in the classical (decoherence) limit. This also leads to a clue about the yet open direction of research in foundational aspects of decoherence of quantum fields and quantum noise. Recently, very interesting observational results [8] over quantum noise in a different context have been obtained, which we touch upon towards the end of this article.

A two point noise kernel for quantum field fluctuations forms a quantity of central importance in the theory of semiclassical stochastic gravity. A semiclassical Einstein Langevin (E-L) equation [7, 9, 10] is based on this quantum noise which forms the source of induced perturbations of the metric in backreaction studies. The semiclassical theory of stochastic gravity is aimed at studying structure formation in the early Universe via the perturbations of the metric and their correlations. The homogeneous solutions of the semiclassical E-L equations describe the intrinsic metric fluctuations, which are same as the solutions of the perturbed Einstein's equation (semiclassical). The inhomogeneous part gives the induced fluctuations of the metric due to the noise. The complete solutions of the semiclassical E-L equation are equivalent to the correlations of quantum perturbations of metric as would be obtained in a viable theory of quantum gravity. The induced metric fluctuations can only be obtained with the term proportional to the noise kernel.

Our aim for studying the correspondence between the fluctuations in the quantum and the classical approximation for matter fields is towards a field-fluid correspondence in the hydrodynamic limit and related areas of active research. The results and connection shown here can thus be useful for the theoretical foundations in field fluid correspondence. Though the E-L formulation aims at exploring the metric perturbations of the spacetime, the modeling of noise in terms of matter field fluctuations has its own importance, apart from the back reaction studies on the geometry. Such a noise kernel would enable a statistical physics analysis for the properties of fluid matter at mesoscopic scales. It is then possible to study, not just the local but also global/extended statistical properties of matter with the two point or higher correlations of the fluctuations in a spacetime structure background. We aim for probing through the sub-hydro mesoscopic scales for the nature of dense matter fluids with this approach in future studies. These scales are expected to lie above the quantum microscopic length scales and the semiclassical mesoscopic scales which the semiclassical theory of stochastic gravity is aimed at studying. We are inclined to address the fluctuations in the decoherence limit, thus moving towards the macroscopic hydrodynamic scales from below. One of the applications of such a result can be aimed at later stages in the evolution of the Universe after the epoch of decoherence of the inflaton field. The hydrodynamic approximation of the matter fluctuations would be useful for the solution of the E-L equation for such an application in cosmology as well.

A new framework has recently started taking shape [5, 11, 12] with which one can expect a few branched out directions ensuing from the basic ideas.

We begin with the stress energy tensor for the scalar field and that for the fluids in standard form.

In this article for the sake of clarity in notation, we denote by ${T}_{ab}^{(\mathrm{fl}\mathrm{u}\mathrm{i}\mathrm{d})}$ and ${N}_{ab{c}^{\prime }{d}^{\prime }}^{(\mathrm{fl}\mathrm{u}\mathrm{i}\mathrm{d})}$ the stress tensor and noise kernel in the fluid approximation, while for the scalar fields we use ${T}_{ab}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})}$ and ${N}_{ab{c}^{\prime }{d}^{\prime }}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})}$.

It is known that a stress tensor for a scalar field can be approximated by a general fluid [1] where,

$$\begin{align}\hfill {T}_{ab}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})}(x)& ={\phi }_{;a}{\phi }_{;b}-\frac{1}{2}{g}_{ab}{\phi }^{;c}{\phi }_{;c}-\frac{1}{2}{g}_{ab}{m}^{2}{\phi }^{2}\hfill \\ \hfill & \quad +\xi ({g}_{ab}\square -{\nabla }_{a}{\nabla }_{b}+{G}_{ab}){\phi }^{2}\hfill \end{align} \tag{ 1 }$$

and with the Klein Gordon field ϕ satisfying

$$\begin{equation}(\square +{m}^{2}+\xi R)\phi =0\end{equation} \tag{ 2 }$$

has a correspondence with

$$\begin{equation}{T}_{ab}^{(\mathrm{fl}\mathrm{u}\mathrm{i}\mathrm{d})}(x)={u}_{\mathrm{a}}{u}_{b}({\epsilon}+p)-{g}_{ab}p+{q}_{a}{u}_{b}+{u}_{\mathrm{a}}{q}_{b}-{\pi }_{ab},\end{equation} \tag{ 3 }$$

where and p are the energy density and pressure of the fluid, u_a the four-velocity and q_a and π_ab the heat flux and anisotropic stress.

For the quantum to classical transition of the system [13, 14]such as in stochastic inflation, the hydrodynamic approximation can be seen to be applicable. Though these correspondences are open to detailed investigations [4], they provide a basis for many well established studies.

The concept of a stochastic process assumes a physical quantity to vary randomly w.r.t. time. However for an underlying general spacetime structure, the notion of time is more involved and specific to the case for a given problem, e.g. a 3 + 1 split, with time orthogonal to the three-dim spatial hypersurface.

Our attempt is to generalize the concept of stochasticity for classical macroscopic variables defined on an underlying spacetime structure. In order to include randomness with respect to the space-time coordinates, rather than just the temporal (or time) coordinate we make this effort. Thus introducing a new terminology of 'generalized stochasticity' enables probabilistic approach in its simplest form (using the first principles) to be extended for physical variables w.r.t. the spacetime coordinates. With this one can address spatial randomness or roughness on a par with stochastic fluctuations (temporal). This enhances the framework of probabilistic and statistical analysis for physical variables on an underlying spacetime structure in relativity. A physical example can be, random variations of pressure or energy density with respect to spatial coordinates in an astrophysical system composed of dense compact matter. It is understood that, such macroscopic quantities like pressure and energy density in any system are a result of smoothened ideal approximations above a certain scale (e.g. a hydrostatic/hydrodynamic scale). In order to probe the scales which are below this, and lie between micro and macro scales in curved spacetime, the generalized stochasticity concept can be useful.

In principle, the transition from macro to micro or vice the versa is expected to have a regime inbetween, which is not very well understood or formulated for various astrophysical systems and compact bodies. Extending the concept of stochasticity in the way we propose here, is useful not just for the conventional processes w.r.t. time, but allows to define 'roughness in physical quantities' which may shed light on interesting properties of exotic matter at intermediate length scales. For example, transport properties and non-equilibrium phenomena in the interiors of dense matter stars can be studied at these intermediate scales with the new theoretical developments.

We define a classical random field (scalar) X[g_ab (x), x) on a spacetime background as X: {xⁱ } → R (similarly vector and tensor random fields can be defined ) where {xⁱ } are coordinates on a pseudo-Reimanian manifold with metric g_ab and a 3 + 1 spacetime split. Note that g_ab metric itself is not to be considered as a random tensor, nor the coordinates {xⁱ } are random variables. The probability distribution of the classical random field is denoted by $\tilde{P}[X]=\int P[X]\mathcal{D}X$, P[X] being its probability density and $\mathcal{D}X$ is a functional integral. Also X would reduce to a regular stochastic field if it were a random function of 't' only. It is a 'generalized stochastic variable' or field having randomness w.r.t. the temporal as well as spatial components of xⁱ . This extension also calls for the possibility that X may show randomness, only w.r.t. spatial coordinates. Such concepts have been used in DDFT theories [18] and give us useful ideas for basic new constructs in our formulation.

Models of noise with the above construct have been used recently in [5, 11, 12] for toy models of perturbed spherically symmetric relativistic stars with stochastic effects. However, it is here in the above section that we have introduced the terminology of 'generalized stochasticity' formally. This may not seem much different from the way quantum scalar field $\hat{\phi }(x)$ and its expectation $\langle \hat{\phi }(x)\rangle =\int \hat{\phi }(x)P[\hat{\phi }]\mathcal{D}\hat{\phi }$ are considered, where $\mathcal{D}\hat{\phi }$ is the functional integral with $\hat{\phi }$ being a quantum operator. The fluctuations of the quantum scalar field as given in equations (6)–(8) (which have been obtained in [6, 7]), follow from the first principles of quantum field theory.

We show the relevance of introducing our framework formally in the following sections.

It is known that fluctuations are inherent to quantum phenomena, while in the decoherence limit the fields become classical. It is expected then that the fluctuations of fields would vanish once they decohere. However, structure formation is known to take place due to these initial quantum fluctuations in the Universe. Though the noise obtained from quantum fluctuations of matter fields has an underlying spacetime on which they are defined, the effect of spatial randomness does not necessarily need an explicit attention. The field fluctuations in cosmology are often addressed in the time domain only, given at a particular spatial coordinate. One may need to address more on this over other applications on qft in curved spacetime. Though the quantum averaging or expectation is done in a regular fashion for observables, there is no discrepancy that our concept of generalized randomness would raise mathematical/technical issues over the well-established semiclassical noise kernel. In this article, we address the classical fluid variables and their randomness, as expected to be seen in the decoherence or classical limit of quantum fields. However, we do not delve into quantum level description of this transition or how it occurs/emerges, but give the results once this transition is complete. This is due to our interest in moving towards coarse grained fluctuations in terms of macroscopic physical fluid variables.

3.1. The semiclassical noise kernel in decoherence limit

The correspondence that we are interested in showing can be obtained by beginning with the exact form of the semiclassical noise kernel as is given in [2, 6] for quantum scalar fields.

This is defined as a bitensor,

$$\begin{equation}8{N}_{ab{c}^{\prime }{d}^{\prime }}(x,{x}^{\prime })=\langle \left\{{\hat{T}}_{ab}(x),{\hat{T}}_{{c}^{\prime }{d}^{\prime }}({x}^{\prime })\right\}\rangle -2\langle {\hat{T}}_{ab}(x)\rangle \langle {\hat{T}}_{{c}^{\prime }{d}^{\prime }}({x}^{\prime })\rangle \end{equation} \tag{ 4 }$$

where ${\hat{T}}_{ab}$ denotes the quantum stress tensor obtained by raising ϕ in (1) to an operator. The expectation of such a quantum stress tensor $\langle {\hat{T}}_{ab}(x)\rangle$ after regularization is used as the matter sector in the semiclassical Einstein's equations. Its fluctuations give a noise kernel (4) which forms the central quantity of importance in the theory of semiclassical stochastic gravity, as mentioned earlier. An elaborate procedure using the point splitting formalism to deal with ill defined operators like ${\hat{\phi }}^{2}$ in the quantum stress tensor in (4) have been used in a straightforward but elaborate way. The noise kernel bitensor is obtained [17] in terms of two point (positive) Wightman functions (as given below) denoted and defined as $G\equiv G(x,{x}^{\prime })=\langle \hat{\phi }(x)\hat{\phi }({x}^{\prime })\rangle$. The details have been worked out in [2, 6], with the final form of the semiclassical noise kernel as,

$$\begin{align}\hfill {N}_{ab{c}^{\prime }{d}^{\prime }}(x,{x}^{\prime })& ={\tilde{N}}_{ab{c}^{\prime }{d}^{\prime }}(x,{x}^{\prime })+{g}_{ab}(x){\tilde{N}}_{{c}^{\prime }{d}^{\prime }}(x,{x}^{\prime })+{g}_{{c}^{\prime }{d}^{\prime }}({x}^{\prime }){\tilde{{N}^{\prime }}}_{ab}(x,{x}^{\prime })\hfill \\ \hfill & \quad +{g}_{ab}(x){g}_{{c}^{\prime }{d}^{\prime }}({x}^{\prime })\tilde{N}(x,{x}^{\prime })\hfill \end{align} \tag{ 5 }$$

with

$$\begin{align}\hfill 8{\tilde{N}}_{ab{c}^{\prime }{d}^{\prime }}(x,{x}^{\prime })& ={(1-2\xi )}^{2}({G}_{;{c}^{\prime }b}{G}_{;{d}^{\prime }a}+{G}_{;{c}^{\prime }a}{G}_{;{d}^{\prime }b})\hfill \\ \hfill & \quad +4{\xi }^{2}({G}_{;{c}^{\prime }{d}^{\prime }}{G}_{;ab}+G{G}_{;ab{c}^{\prime }{d}^{\prime }})-2\xi (1-2\xi )\left({G}_{;b}{G}_{;{c}^{\prime }a{d}^{\prime }}\right.\hfill \\ \hfill & \quad \left.+{G}_{;a}{G}_{;{c}^{\prime }b{d}^{\prime }}\left.+{G}_{;{d}^{\prime }}{G}_{;ab{c}^{\prime }}+{G}_{;{c}^{\prime }}{G}_{;ab{d}^{\prime }}\right)\right.\hfill \\ \hfill & \quad +2\xi (1-2\xi )\left({G}_{;a}{G}_{;b}{R}_{{c}^{\prime }{d}^{\prime }}\left.+{G}_{;{c}^{\prime }}{G}_{;{d}^{\prime }}{R}_{ab}\right)\right.\hfill \\ \hfill & \quad -4{\xi }^{2}({G}_{;ab}{R}_{{c}^{\prime }{d}^{\prime }}+{G}_{;{c}^{\prime }{d}^{\prime }}{R}_{ab})G+2{\xi }^{2}{R}_{{c}^{\prime }{d}^{\prime }}{R}_{ab}{G}^{2}\hfill \end{align} \tag{ 6 }$$

$$\begin{align}\hfill \quad 8{\tilde{N}}_{ab}^{\prime }(x,{x}^{\prime })& =2(1-2\xi )\left[\left(2\xi -\frac{1}{2}\right){G}_{;{p}^{\prime }b}{{{G}_{;}}^{{p}^{\prime }}}_{a}+\xi \left({G}_{;b}{{G}_{;{p}^{\prime }a}}^{{p}^{\prime }}\right.\right.\hfill \\ \hfill & \quad \left.\left.+G{;}_{a}{{G}_{;{{p}^{\prime }}_{b}}}^{{p}^{\prime }}\right)\right]-4\xi \left[\left(2\xi -\frac{1}{2}\right){{G}_{;}}^{{p}^{\prime }}{G}_{;ab{p}^{\prime }}\right.\hfill \\ \hfill & \quad \left.+\xi ({G}_{;{p}^{\prime }}^{{\enspace p}^{\prime }}{G}_{;ab}+G{{G}_{;ab{p}^{\prime }}}^{{p}^{\prime }})\right]-({m}^{2}+\xi {R}^{\prime })\left[(1-2\xi ){G}_{;a}{G}_{;b}\right.\hfill \\ \hfill & \quad \left.-2G\xi {G}_{;ab}\right]+2\xi \left[\left(2\xi -\frac{1}{2}\right){G}_{;{p}^{\prime }}{{G}_{;}}^{{p}^{\prime }}\right.\hfill \\ \hfill & \quad \left.+2G\xi {{G}_{;{p}^{\prime }}}^{{p}^{\prime }}\right]{R}_{ab}-({m}^{2}+\xi {R}^{\prime })\xi {R}_{ab}{G}^{2}\hfill \end{align} \tag{ 7 }$$

$$\begin{align}\hfill \quad \quad \quad \quad 8\tilde{N}& =2{\left(2\xi -\frac{1}{2}\right)}^{2}{{G}_{;{p}^{\prime }}}_{q}{G}_{;}^{{p}^{\prime }q}+4{\xi }^{2}({{G}_{;{p}^{\prime }}}^{{p}^{\prime }}{G}_{;q}^{q}+G{{{{G}_{;p}}^{p}}_{{q}^{\prime }}}^{{q}^{\prime }})\hfill \\ \hfill & \quad +4\xi \left(2\xi -\frac{1}{2}\right)({G}_{;p}{{G}_{;{q}^{\prime }}}^{{pq}^{\prime }}+{{G}_{;}}^{{p}^{\prime }}{{G}_{;q}}_{{p}^{\prime }}^{q})\hfill \\ \hfill & \quad -\left(2\xi -\frac{1}{2}\right)[({m}^{2}+\xi R){G}_{;{p}^{\prime }}{G}_{;}^{{p}^{\prime }}+({m}^{2}+\xi {R}^{\prime }){G}_{;p}{G}_{;}^{p}]\hfill \\ \hfill & \quad -2\xi [({m}^{2}+\xi R){{G}_{;{p}^{\prime }}}^{{p}^{\prime }}+({m}^{2}+\xi {R}^{\prime }){{G}_{;p}}^{p}]G\hfill \\ \hfill & \quad +\frac{1}{2}({m}^{2}+\xi R)({m}^{2}+\xi {R}^{\prime }){G}^{2}\hfill \end{align} \tag{ 8 }$$

The quantity ${\tilde{N}}_{{c}^{\prime }{d}^{\prime }}$ in equation (5) can be easily obtained from equation (6) by taking ${g}^{ab}{\tilde{N}}_{ab{c}^{\prime }{d}^{\prime }}\to {{\tilde{N}}^{a}}_{a{c}^{\prime }{d}^{\prime }}={\tilde{N}}_{{c}^{\prime }{d}^{\prime }}$, using the properties satisfied by the noise kernel as discussed with all details in [7].

Now, for this noise kernel the decoherence limit can be taken simply with the quantum stress tensor in (4) replaced by the classical stress tensor, which then takes the form

$$\begin{align}\hfill 8{N}_{ab{c}^{\prime }{d}^{\prime }}(x,{x}^{\prime })& =2(\langle {T}_{ab}(x){T}_{{c}^{\prime }{d}^{\prime }}({x}^{\prime })\rangle -\langle {T}_{ab}(x)\rangle \langle {T}_{{c}^{\prime }{d}^{\prime }}({x}^{\prime })\rangle )\hfill \\ \hfill & =2\,\text{Cov}[{T}_{ab}(x){T}_{{c}^{\prime }{d}^{\prime }}({x}^{\prime })]\hfill \end{align} \tag{ 9 }$$

The above form is common for the classical fields and the fluids in terms of the stress tensor T_ab , hence we do not put labels for the nature of noise and stress tensors in this expression in terms of field or fluid.

In the decoherence limit, the stress tensor can be said to contain the classical scalar field ϕ with a mesoscopic scale randomness, giving it a probability distribution. The averages in the classical limit are then the statistical averages, where the classical scalar field expectation is $\langle \phi (x)\rangle =\int \phi (x)P[\phi ]\mathcal{D}\phi$. Then the Wightman functions reduce to the two point form $\langle \phi (x)\phi ({x}^{\prime })\rangle =\int \phi (x)\phi ({x}^{\prime })P[\phi ]\mathcal{D}\phi$. We connect with this, the fluid approximation in the section below.

3.2. Scalar field and fluid approximation with statistical averages

Following on the lines of [1], the relation between the stress tensors for scalar fields and fluids is given by

$$\begin{equation}{T}_{ab}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})}={(1-\xi {\phi }^{2})}^{-1}[{S}_{ab}+\xi \left\{{g}_{ab}\square ({\phi }^{2})-{\nabla }_{a}{\nabla }_{b}({\phi }^{2})\right\}]\end{equation} \tag{ 10 }$$

where

$$\begin{equation*}{S}_{ab}={\partial }_{a}\phi {\partial }_{b}\phi -{\mathcal{L}}_{\phi }{g}_{ab}\end{equation*}$$

with

$$\begin{equation*}{\mathcal{L}}_{\phi }=\frac{1}{2}{\partial }_{a}\phi {\partial }^{a}\phi -V[\phi ]\end{equation*}$$

in what follows, we consider a specific form of the potential $V(\phi )=-\frac{1}{2}{m}^{2}\phi$. However, equivalent treatment to other cases can be given easily. The corresponding effective fluid stress tensor is then given by [1]

$$\begin{equation}{T}_{ab}^{(\mathrm{fl}\mathrm{u}\mathrm{i}\mathrm{d})}={u}_{\mathrm{a}}{u}_{b}(p+{\epsilon})-{g}_{ab}p+{q}_{a}{u}_{b}+{u}_{\mathrm{a}}{q}_{b}-{\pi }_{ab}\end{equation} \tag{ 11 }$$

such that

$$\begin{equation}{u}_{\mathrm{a}}={[{\partial }_{c}(\phi ){\partial }^{c}(\phi )]}^{-1/2}{\partial }_{a}\phi \end{equation} \tag{ 12 }$$

$$\begin{align}\hfill \ {\epsilon}& ={(1-\xi {\phi }^{2})}^{-1}\left[\frac{1}{2}{\partial }_{c}\phi {\partial }^{c}\phi +V(\phi )+\xi \left\{\square ({\phi }^{2})\right.\right.\left.\left.-{({\partial }^{c}\phi {\partial }_{c}\phi )}^{-1}{\partial }^{a}\phi {\partial }^{b}\phi {\nabla }_{a}{\nabla }_{b}({\phi }^{2})\right\}\right]\hfill \end{align} \tag{ 13 }$$

$$\begin{equation}{q}_{a}=\xi {(1-\xi {\phi }^{2})}^{-1}{({\partial }^{b}\phi {\partial }_{b}\phi )}^{-3/2}{\partial }^{c}\phi {\partial }^{d}\phi [{\nabla }_{c}{\nabla }_{d}({\phi }^{2}){\partial }_{a}\phi -{\nabla }_{a}{\nabla }_{c}({\phi }^{2}){\partial }_{d}\phi ]\end{equation} \tag{ 14 }$$

$$\begin{align}\hfill \ p& ={(1-\xi {\phi }^{2})}^{-1}\left[\frac{1}{2}{\partial }_{c}\phi {\partial }^{c}\phi -V(\phi )-\xi \right.\hfill \\ \hfill & \quad \times \left.\left\{\frac{2}{3}\square ({\phi }^{2})+\frac{1}{3}{({\partial }_{c}\phi {\partial }^{c}\phi )}^{-1}{\nabla }_{a}{\nabla }_{b}({\phi }^{2}){\partial }^{a}\phi {\partial }^{b}\phi \right\}\right]\hfill \end{align} \tag{ 15 }$$

$$\begin{align}\hfill {\pi }_{ab}& =\xi {(1-\xi {\phi }^{2})}^{-1}{({\partial }^{c}\phi {\partial }_{c}\phi )}^{-1}\left[\frac{1}{3}({\partial }_{a}\phi {\partial }_{b}\phi -{g}_{ab}{\partial }^{d}\phi {\partial }_{d}\phi )\right.\hfill \\ \hfill & \quad \times \left.\left\{\square ({\phi }^{2})-{({\partial }^{d}\phi {\partial }_{d}\phi )}^{-1}{\nabla }_{l}{\nabla }_{m}({\phi }^{2}){\partial }^{l}\phi {\partial }^{m}\phi \right\}\right.\hfill \\ \hfill & \quad \left.+{\partial }^{l}\phi \left\{{\nabla }_{a}{\nabla }_{b}({\phi }^{2}){\partial }_{l}\phi -{\nabla }_{a}{\nabla }_{l}({\phi }^{2}){\partial }_{b}\phi -{\nabla }_{l}{\nabla }_{b}({\phi }^{2}){\partial }_{a}\phi \right.\right.\hfill \\ \hfill & \quad \left.\left.+{({\partial }_{p}\phi {\partial }^{p}\phi )}^{-1}{\partial }^{m}\phi {\nabla }_{m}{\nabla }_{l}({\phi }^{2}){\partial }_{a}\phi {\partial }_{b}\phi \right\}\right]\hfill \end{align} \tag{ 16 }$$

For our work, we can assign expectation values to the scalar field terms in the above expression as mentioned earlier. The random fluctuations that we intend to focus on can then be defined as δ_R ϕ(x) = ϕ(x) − 〈ϕ(x)〉, where δ_R stands for the random fluctuations such that $\langle {\delta }_{\mathrm{R}}\phi (x)\rangle =0$. The picture of the effective fluid has subtle differences with the regular fluids. This is a topic of active interest in research, with many details still open to investigations [13, 19]. Thus the fluctuations at mesoscopic scales that we talk about here, are not necessarily to be associated with thermal fluctuations in a regular fluid or with Brownian motion of particles. The concept of effective fluid is thus much more expansive, extending the hydro approximation to different forms of matter. The microscopic and mesoscopic level description in such a case may be very different than the ordinary fluids.

Often for a stochastic effect to be realised in a physical system, a kinematic variable is considered. In general, for the case of a relativistic fluid one expects the fluid four-velocity to carry any stochastic effects in the hydrodynamic description. However, for an effective fluid, as we consider here, one can see from equation (12) that the definition of four-velocity u_a is a normalized four-vector with a unit modulus. For any randomness in the fluid though its modulus remains unchanged, its direction which is normal to the hypersurfaces of constant $\hat{\phi }$, can vary randomly for different realizations of the stochastic field $\hat{\phi }(x)$. Such a picture would hold for a quantum scalar field surely where this is possible due to coherence of states. But once the states decohere and the scalar field becomes classical, it is only the magnitude which can change in different realizations, while the underlying geometry is deterministic. We are interested in taking the decoherence limit of the quantum scalar field for our work. Thus we assume that in this limit, the operator valued quantum field collapses into a classical scalar field state. It is known that the decoherence phenomena does not lead to complete vanishing of the phases, but one can consider a classical picture of the fields, with the assumption that the randomness w.r.t. the direction of the four-velocity is negligible. One way is to move from the quantum scalar fields to the classical picture first and then define the four-velocity in the hydrodynamic limit. The second approximation which one may be able to consider, is that, even without losing the quantum nature, the fluctuations in the directions of the unit velocity vector (which is the kinetic term), are negligible compared to the 'bulk properties' which fluctuate with dependence on the magnitude of fields. An example is, while considering squeezing of states, one can make the phases negligible and magnitude fluctuations higher, which is allowed by the uncertainty principle. However the correspondence that we work out here, is specifically for the decoherence limit of fields which is closer to the former case. But it would be interesting to explore more about squeezed states in this fashion and try to form such a correspondence with the fluid approximation. Thus in this article we consider u_a as a deterministic quantity where the randomness is negligible (similar to [5]).

3.3. The generalized noise in the system

The effort that we are making here, is useful for capturing 'partial microscopic' or mesoscopic scale effects in a classical description of the system. Any externally induced mechanical effects which are random in nature can also cause such fluctuations/roughness in physical parameters at different scales in the system.

We see that knowing the analytical form of the semiclassical noise kernel makes it possible for us to find the correspondence that we are interested in. We incorporate the decoherence limit of the scalar fields with further extension to the fluid approximation. Thus we obtain our results in terms of two point fluctuations of fluid variables as shown below.

The effective noise kernel in terms of the non-ideal fluid variables can be obtained by using (11) in (9) as follows ('Cov' in the expressions below denotes statistical covariance of quantities as defined in equation (9))

$$\begin{align}\hfill 4{N}_{ab{c}^{\prime }{d}^{\prime }}^{(\mathrm{fl}\mathrm{u}\mathrm{i}\mathrm{d})}(x,{x}^{\prime })& =\text{Cov}[{T}_{ab}^{(\mathrm{fl}\mathrm{u}\mathrm{i}\mathrm{d})}(x){T}_{{c}^{\prime }{d}^{\prime }}^{(\mathrm{fl}\mathrm{u}\mathrm{i}\mathrm{d})}({x}^{\prime })]={u}_{\mathrm{a}}(x){u}_{b}(x){u}_{{c}^{\prime }}({x}^{\prime }){u}_{{d}^{\prime }}\hfill \\ \hfill & \quad \times ({x}^{\prime })\left\{\text{Cov}[{\epsilon}(x){\epsilon}({x}^{\prime })]+\text{Cov}[{\epsilon}(x)p({x}^{\prime })]\right.\left.+\text{Cov}[p(x){\epsilon}({x}^{\prime })]+\text{Cov}[p(x)p({x}^{\prime })]\right\}\hfill \\ \hfill & \quad -{u}_{\mathrm{a}}(x){u}_{b}(x){g}_{{c}^{\prime }{d}^{\prime }}({x}^{\prime })\left\{\text{Cov}[{\epsilon}(x)p({x}^{\prime })]\right.\left.+\text{Cov}[p(x)p({x}^{\prime })]\right\}-{g}_{ab}(x){u}_{{c}^{\prime }}({x}^{\prime }){u}_{{d}^{\prime }}({x}^{\prime })\hfill \\ \hfill & \quad \times \left\{\text{Cov}[p(x){\epsilon}({x}^{\prime })]+\text{Cov}[p(x)p({x}^{\prime })]\right\}+{g}_{ab}(x){g}_{{c}^{\prime }{d}^{\prime }}({x}^{\prime })\text{Cov}[p(x)p({x}^{\prime })]\hfill \\ \hfill & \quad +{u}_{\mathrm{a}}(x){u}_{{c}^{\prime }}({x}^{\prime })\text{Cov}[{q}_{b}(x),{q}_{{d}^{\prime }}({x}^{\prime })]+{u}_{\mathrm{a}}(x){u}_{{d}^{\prime }}({x}^{\prime })\text{Cov}[{q}_{b}(x){q}_{{c}^{\prime }}({x}^{\prime })]\hfill \\ \hfill & \quad +{u}_{b}(x){u}_{{c}^{\prime }}({x}^{\prime })\text{Cov}[{q}_{a}(x){q}_{{d}^{\prime }}({x}^{\prime })]+{u}_{b}(x){u}_{{d}^{\prime }}({x}^{\prime })\text{Cov}[{q}_{a}(x){q}_{{c}^{\prime }}({x}^{\prime })]\hfill \\ \hfill & \quad +\text{Cov}[{\pi }_{ab}(x){\pi }_{{c}^{\prime }{d}^{\prime }}({x}^{\prime })]\hfill \end{align} \tag{ 17 }$$

where the two point covariance Cov[p(x), p(x')] = 〈p(x)p(x')〉 − 〈p(x)〉〈p(x')〉 for pressures at two separate spacetime points x and x' are defined. Similar terms for other fluid variables can be seen in the above expression. All the fluid variables showing generalized stochastic nature are defined through the common form in the curved spacetime, e.g. pressure given by $\langle p(x)\rangle =\int p(x)P[p]\mathcal{D}p$, while $\langle p(x)p({x}^{\prime })\rangle =\int p(x)p({x}^{\prime })P[p]\mathcal{D}p$. To be more specific p ≡ p[g_ab , x) for all our considerations, with similar forms for the rest of the fluid variables. We can compare the above expression with the form of equation (5) of the semiclassical case, expressed in terms of coefficients of the spacetime metric g_ab . The metric is common to the semiclassical and classical case.

Comparing equations (5)–(8) with equation (17) it is straightforward to relate terms which are coefficients of g_ab (x), g_cd (x'), g_ab (x)g_cd (x') as corresponding (the sign → in the equations below and further in this article is used to denote 'term corresponds to') in the two cases to get,

$$\begin{align}\hfill & 8{\tilde{N}}_{ab{c}^{\prime }{d}^{\prime }}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})}\to 8{N}_{ab{c}^{\prime }{d}^{\prime }}^{(\mathrm{fl}\mathrm{u}\mathrm{i}\mathrm{d})}\qquad (\text{relating}\enspace \text{the corresponding terms as}\;\text{:})\;\hfill \\ \hfill & \left\{{(1-2\xi )}^{2}({G}_{;{c}^{\prime }b}{G}_{;{d}^{\prime }a}+{G}_{;{c}^{\prime }a}{G}_{;{d}^{\prime }b})+4{\xi }^{2}({G}_{;{c}^{\prime }{d}^{\prime }}{G}_{;ab}+G{G}_{;ab{c}^{\prime }{d}^{\prime }})\right.\hfill \\ \hfill & \quad \left.-2\xi (1-2\xi )\left({G}_{;b}{G}_{;{c}^{\prime }a{d}^{\prime }}+{G}_{;a}{G}_{;{c}^{\prime }b{d}^{\prime }}\right.\left.+{G}_{;{d}^{\prime }}{G}_{;ab{c}^{\prime }}+{G}_{;{c}^{\prime }}{G}_{;ab{d}^{\prime }}\right)\right.\hfill \\ \hfill & \quad \left.+2\xi (1-2\xi )({G}_{;a}{G}_{;b}{R}_{{c}^{\prime }{d}^{\prime }}+{G}_{;{c}^{\prime }}{G}_{;{d}^{\prime }}{R}_{ab})\right.\left.-4{\xi }^{2}({G}_{;ab}{R}_{{c}^{\prime }{d}^{\prime }}+{G}_{;{c}^{\prime }{d}^{\prime }}{R}_{ab})G\right.\hfill \\ \hfill & \quad \left.+2{\xi }^{2}{R}_{{c}^{\prime }{d}^{\prime }}{R}_{ab}{G}^{2}\right\}\to \left\{2\left\{{u}_{\mathrm{a}}(x){u}_{b}(x){u}_{{c}^{\prime }}({x}^{\prime }){u}_{{d}^{\prime }}({x}^{\prime })\left\{\text{Cov}[{\epsilon}(x){\epsilon}({x}^{\prime })]\right.\right.\right.\hfill \\ \hfill & \quad \left.\left.\left.+\text{Cov}[{\epsilon}(x)p({x}^{\prime })]+\text{Cov}[p(x){\epsilon}({x}^{\prime })]+\text{Cov}[p(x)p({x}^{\prime })]\right\}\right.\right.\hfill \\ \hfill & \quad \left.\left.+{u}_{\mathrm{a}}(x){u}_{{c}^{\prime }}({x}^{\prime })\text{Cov}[{q}_{b}(x),{q}_{{d}^{\prime }}({x}^{\prime })]+{u}_{\mathrm{a}}(x){u}_{{d}^{\prime }}({x}^{\prime })\text{Cov}[{q}_{b}(x){q}_{{c}^{\prime }}({x}^{\prime })]\right.\right.\hfill \\ \hfill & \quad \left.\left.+{u}_{b}(x){u}_{{c}^{\prime }}({x}^{\prime })\text{Cov}[{q}_{a}(x){q}_{{d}^{\prime }}({x}^{\prime })]\right.\right.\hfill \\ \hfill & \quad \left.\left.+{u}_{b}(x){u}_{{d}^{\prime }}({x}^{\prime })\text{Cov}[{q}_{a}(x){q}_{{c}^{\prime }}({x}^{\prime })]+\text{Cov}\left[\right.{\pi }_{ab}(x){\pi }_{{c}^{\prime }{d}^{\prime }}({x}^{\prime })\right\}\right\}\hfill \end{align} \tag{ 18 }$$

$$\begin{align}\hfill 8{\tilde{{N}^{\prime }}}_{ab}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})}& \to 8{{N}^{\prime }}_{ab}^{(\mathrm{fl}\mathrm{u}\mathrm{i}\mathrm{d})}\qquad (\text{relating the corresponding terms as}\;\text{:})\;\hfill \\ \hfill & \quad \times \left\{2(1-2\xi )\left[\left(2\xi -\frac{1}{2}\right){G}_{;{p}^{\prime }b}{{{G}_{;}}^{{p}^{\prime }}}_{a}+\xi \left({G}_{;b}{G}_{;{p}^{\prime }a}^{{\quad p}^{\prime }}\right.\right.\right.\hfill \\ \hfill & \quad \left.\left.\left.+{G}_{;a}{G}_{;{p}^{\prime }b}^{{\quad p}^{\prime }}\right)\right]-4\xi \left[\left(2\xi -\frac{1}{2}\right){{G}_{;}}^{{p}^{\prime }}{G}_{;ab{p}^{\prime }}+\xi \left({{G}_{;{p}^{\prime }}}^{{p}^{\prime }}{G}_{;ab}\right.\right.\right.\hfill \\ \hfill & \quad \left.\left.\left.+G{{G}_{;ab{p}^{\prime }}}^{{p}^{\prime }}\right)\right]\right.\left.-({m}^{2}+\xi {R}^{\prime })[(1-2\xi ){G}_{;a}{G}_{;b}-2G\xi {G}_{;ab}]\right.\hfill \\ \hfill & \quad \left.+2\xi \left[\left(2\xi -\frac{1}{2}\right){G}_{;{p}^{\prime }}{{G}_{;}}^{{p}^{\prime }}+2G\xi {{G}_{;{p}^{\prime }}}^{{p}^{\prime }}\right]{R}_{ab}\left.-({m}^{2}+\xi {R}^{\prime })\xi {R}_{ab}{G}^{2}\right\}\right.\hfill \\ \hfill & \quad \to \left\{-2{u}_{\mathrm{a}}(x){u}_{b}(x)\left\{\text{Cov}[{\epsilon}(x)p({x}^{\prime })]\right.\right.\left.\left.+\text{Cov}[p(x)p({x}^{\prime })]\right\}\right\}\hfill \end{align} \tag{ 19 }$$

$$\begin{align}\hfill 8{\tilde{N}}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})}& \to 8{N}^{(\mathrm{fl}\mathrm{u}\mathrm{i}\mathrm{d})}\qquad (\text{relating the corresponding terms as}\;\text{:})\;\hfill \\ \hfill & \quad \times \left\{2{\left(2\xi -\frac{1}{2}\right)}^{2}{{G}_{;{p}^{\prime }}}_{q}{{G}_{;}}^{{p}^{\prime }q}+4{\xi }^{2}({{G}_{;{p}^{\prime }}}^{{p}^{\prime }}{{G}_{;q}}^{q}+G{{{G}_{;p}}^{p}}_{{ q}^{\prime }}^{\enspace {q}^{\prime }})\right.\hfill \\ \hfill & \quad \left.+4\xi \left(2\xi -\frac{1}{2}\right)\left({G}_{;p}{{G}_{;{q}^{\prime }}}^{p{q}^{\prime }}\right.\left.+{G}_{{;}^{{p}^{\prime }}}{{G}_{;q}}_{{p}^{\prime }}^{q}\right)\right.\hfill \\ \hfill & \quad \left.-\left(2\xi -\frac{1}{2}\right)\left[({m}^{2}+\xi R){G}_{;{p}^{\prime }}{G}_{{;}^{{p}^{\prime }}}\right.\right.\left.\left.+({m}^{2}+\xi {R}^{\prime }){G}_{;p}{G}_{{;}^{p}}\right]\right.\hfill \\ \hfill & \quad \left.-2\xi [({m}^{2}+\xi R){G}_{;}{{p}^{\prime }}^{{p}^{\prime }}+({m}^{2}+\xi {R}^{\prime }){G}_{;{p}^{p}}]G\right.\hfill \\ \hfill & \quad \left.+\frac{1}{2}({m}^{2}+\xi R)({m}^{2}+\xi {R}^{\prime }){G}^{2}\right\}\to \left\{2\text{Cov}[p(x)p({x}^{\prime })]\left.\right]\right\}\hfill \end{align} \tag{ 20 }$$

$$\begin{align}\hfill 8{\tilde{N}}_{{c}^{\prime }{d}^{\prime }}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})}& \to 8{N}_{{c}^{\prime }{d}^{\prime }}^{(\mathrm{fl}\mathrm{u}\mathrm{i}\mathrm{d})}(x,{x}^{\prime })(\text{relating the corresponding terms as}\;\text{:})\;\hfill \\ \hfill & \left\{2{(1-2\xi )}^{2}{{G}_{;{c}^{\prime }}}_{p}{{G}_{;{d}^{\prime }}}^{p}+4{\xi }^{2}({{G}_{;{c}^{\prime }}}_{{d}^{\prime }}{\square }_{x}G+G{\square }_{x}{{G}_{;{c}^{\prime }}}_{{d}^{\prime }})\right.\hfill \\ \hfill & \quad \left.-2\xi (1-2\xi )\left(2{G}_{;p}{{G}_{;{c}^{\prime }}}_{{d}^{\prime }}^{p}\right.\right.\hfill \\ \hfill & \quad \left.\left.+{G}_{;{d}^{\prime }}{\square }_{x}{G}_{;{c}^{\prime }}+{G}_{;{c}^{\prime }}{\square }_{x}{G}_{;{d}^{\prime }}\right)\right.\hfill \\ \hfill & \quad \left.+2\xi (1-2\xi )({G}_{;p}{{G}_{;}}^{p}{R}_{{c}^{\prime }{d}^{\prime }}+{G}_{;{c}^{\prime }}{G}_{;{d}^{\prime }}R)\right.\hfill \\ \hfill & \quad \left.-4{\xi }^{2}({\square }_{x}G{R}_{{c}^{\prime }{d}^{\prime }}+{{G}_{;{c}^{\prime }}}_{{d}^{\prime }}R)G+2{\xi }^{2}{R}_{{c}^{\prime }{d}^{\prime }}R{G}^{2}\right\}\hfill \\ \hfill & \to \left\{-2{u}_{{c}^{\prime }}{u}_{{d}^{\prime }}\left\{\right.\text{Cov}[p(x){\epsilon}({x}^{\prime })]+\text{Cov}[p(x)p({x}^{\prime })]\right\}\hfill \end{align} \tag{ 21 }$$

In the decoherence limit the Wightman function defined for operator valued quantum scalar field $\hat{\phi }$, reduces to two point correlations of the classical scalar field ϕ, such that in the above expressions $G\equiv G(x,{x}^{\prime })\to \int \phi (x)\phi ({x}^{\prime })P[\phi ]\mathcal{D}\phi$ as has been mentioned earlier. The above equations, showing the correspondences in terms of two point functions and fluctuations of fluid variables are the main result that we intend to present here. We see, that the two point covariances of fluid variables, namely pressure, energy density, heat flux and anisotropic stresses, capture the point separated mesoscopic nature of the fluid in a statistical description, which are in turn related to fluctuations of the scalar field. Note that these covariances are generalized w.r.t. the spacetime variables, hence statistically capture the spatial randomness along with the temporal fluctuations. To probe for the sub-hydro mesoscopic scales in the effective fluid theoretically, this is one of the important new developments. The usefulness of this is described elaborately in the next section. In order to extract the covariances of fluid variables explicitly in terms of the field fluctuations, we re-arrange the above set of equations as,

$$\begin{equation}\text{Cov}[p(x)p({x}^{\prime })]=4{\tilde{N}}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})}\end{equation} \tag{ 22 }$$

$$\begin{equation}\text{Cov}[{\epsilon}(x)p({x}^{\prime })]=-\frac{4{{\tilde{N}}^{\prime }}_{ab}{}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})}}{{\partial }_{a}\phi {\partial }_{b}\phi \left.\right)}({\partial }_{e}\phi {\partial }^{e}\phi )-4{\tilde{N}}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})}\end{equation} \tag{ 23 }$$

$$\begin{equation}\text{Cov}[p(x){\epsilon}({x}^{\prime })]=-\frac{4{\tilde{N}}_{{c}^{\prime }{d}^{\prime }}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})}}{{\partial }_{{c}^{\prime }}\phi {\partial }_{{d}^{\prime }}\phi }({\partial }_{e}\phi {\partial }^{e}\phi )-4{\tilde{N}}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})}\end{equation} \tag{ 24 }$$

$$\begin{align}\hfill \text{Cov}[{\epsilon}(x){\epsilon}({x}^{\prime })]+{u}_{\left(\right.a}{u}_{\left(\right.{c}^{\prime }}\text{Cov}\left[\right.{q}_{b\left.\right)}{q}_{{d}^{\prime }\left.\right)} \left.\right]& +\text{Cov}[{\pi }_{ab}{\pi }_{{c}^{\prime }{d}^{\prime }}]=\frac{4{\tilde{N}}_{ab{c}^{\prime }{d}^{\prime }}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})}}{{\partial }_{a}\phi {\partial }_{b}\phi {\partial }_{{c}^{\prime }}\phi {\partial }_{{d}^{\prime }}\phi }{({\partial }_{e}\phi {\partial }^{e})}^{2}\hfill \\ \hfill & +\frac{4{\tilde{{N}^{\prime }}}_{ab}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})}}{{\partial }_{a}\phi {\partial }_{b}\phi }({\partial }_{e}\phi {\partial }^{e}\phi )+\frac{4{\tilde{N}}_{{c}^{\prime }{d}^{\prime }}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})}}{{\partial }_{{c}^{\prime }}\phi {\partial }_{{d}^{\prime }}\phi }({\partial }_{e}\phi {\partial }^{e}\phi )+4{\tilde{N}}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})}\hfill \end{align} \tag{ 25 }$$

where ${\tilde{N}}_{ab{c}^{\prime }{d}^{\prime }}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})},{\tilde{{N}^{\prime }}}_{ab}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})},{\tilde{N}}_{{c}^{\prime }{d}^{\prime }}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})}$ and ${\tilde{N}}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})}$ are given by lhs of equations (18)–(21) in terms of the covariant derivatives of the Wightman functions. On the rhs of the above equation we have used the relation ${u}_{\mathrm{a}}={({\partial }_{c}\phi {\partial }^{c}\phi )}^{-1/2}{\partial }_{a}\phi$, to get our expressions in a consistent form in terms of scalar field only.

4.1. Relations for the effective ideal fluid approximation

The case of ideal or perfect fluid approximation and its fluctuations has been discussed in [5], where we have obtained the correspondences equivalent to expressions (18)–(20). Revising the semiclassical noise kernel when a perfect fluid approximation can be assumed to hold, so that ξ = 0 in all the equations starting from (1), gives

$$\begin{align} 8{\tilde{N}}_{ab{c}^{\prime }{d}^{\prime }}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})}\hfill & \hfill \to 8{N}_{ab{c}^{\prime }{d}^{\prime }}^{(\mathrm{fl}\mathrm{u}\mathrm{i}\mathrm{d})}\qquad (\text{relating the corresponding terms as:})\\ \hfill \left\{{G}_{;{c}^{\prime }b}{G}_{;{d}^{\prime }a}+{G}_{;{c}^{\prime }a}{G}_{;{d}^{\prime }b}\right\}& \to \left\{2{u}_{\mathrm{a}}(x){u}_{b}(x){u}_{c}({x}^{\prime }){u}_{d}({x}^{\prime })\left\{\text{Cov}[{\epsilon}(x){\epsilon}({x}^{\prime })]\right.\right.\left.\left.+\text{Cov}[{\epsilon}(x)p({x}^{\prime })]\right.\right.\hfill \\ \hfill & \quad \left.\left.+\text{Cov}[p(x){\epsilon}({x}^{\prime })]+\text{Cov}[p(x)p({x}^{\prime })]\right\}\right\}\hfill \end{align} \tag{ 26 }$$

$$\begin{align}\hfill & \hfill \\ 8{\tilde{{N}^{\prime }}}_{ab}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})}\hfill & \hfill \to 8{{N}^{\prime }}_{ab}^{(\mathrm{fl}\mathrm{u}\mathrm{i}\mathrm{d})}\qquad (\text{relating the corresponding terms as:})\\ \hfill \left\{-{G}_{;{p}^{\prime }b}{G}_{;\enspace a}^{{\enspace p}^{\prime }}-{m}^{2}{G}_{;a}{G}_{;b}\right\}& \to \left\{-2{u}_{\mathrm{a}}(x){u}_{b}(x)\left\{\text{Cov}[{\epsilon}(x)p({x}^{\prime })]\right.\right.\left.\left.+\text{Cov}[p(x)p({x}^{\prime })]\right\}\right\}\hfill \end{align} \tag{ 27 }$$

$$\begin{align}\hfill & \hfill \\ 8{\tilde{N}}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})}\to 8{N}^{(\mathrm{fl}\mathrm{u}\mathrm{i}\mathrm{d})}\hfill & \quad (\text{relating the corresponding terms as}\;\text{:})\hfill \\ \hfill \left\{\frac{1}{2}{G}_{;{p}^{\prime }q}{G}_{;}^{{\enspace p}^{\prime }q}+\frac{1}{2}{m}^{2}[{G}_{;{p}^{\prime }}{G}_{;}^{{\enspace p}^{\prime }}+{G}_{;p}{G}_{;}^{\enspace p}+{m}^{2}{G}^{2}]\right\}& \to \left\{2\text{Cov}[p(x)p({x}^{\prime })]\right\}\hfill \end{align} \tag{ 28 }$$

$$\begin{align}\hfill & \hfill \\ \hfill 8{\tilde{N}}_{{c}^{\prime }{d}^{\prime }}^{(\mathrm{fi}\mathrm{e}\mathrm{l}\mathrm{d})}& \to 8{N}_{{c}^{\prime }{d}^{\prime }}^{(\mathrm{fl}\mathrm{u}\mathrm{i}\mathrm{d})}\qquad (\text{relating the corresponding terms as}\;\text{:})\hfill \\ \hfill \left\{2{G}_{;{c}^{\prime }p}{{G}_{;{d}^{\prime }}}^{p}\right\}& \to \left\{-2{u}_{{c}^{\prime }}{u}_{{d}^{\prime }}\left\{\text{Cov}[p(x){\epsilon}({x}^{\prime })]+\text{Cov}[p(x)p({x}^{\prime })]\right\}\right\}\hfill \end{align} \tag{ 29 }$$

The reverse equations can be obtained easily from the above in the following form

$$\begin{equation}\text{Cov}[p(x)p({x}^{\prime })]=\frac{1}{4}{G}_{;{p}^{\prime }q}{G}_{;}^{{\enspace p}^{\prime }q}+\frac{1}{4}{m}^{2}[{G}_{;{p}^{\prime }}{G}_{;}^{{\enspace p}^{\prime }}+{G}_{;p}{G}_{;}^{\enspace p}+{m}^{2}{G}^{2}]\end{equation} \tag{ 30 }$$

$$\begin{align}\hfill \text{Cov}[{\epsilon}(x)p({x}^{\prime })]& =\frac{1}{2}({{G}_{;{p}^{\prime }}}_{b}{{G}_{;}}_{\enspace a}^{{p}^{\prime }}+{m}^{2}{G}_{;a}{G}_{;b})\frac{{\partial }^{e}\phi {\partial }_{e}\phi }{{\partial }_{a}\phi {\partial }_{b}\phi }-\frac{1}{4}{{G}_{;{p}^{\prime }}}_{q}{{G}_{;}}^{{p}^{\prime }q}\hfill \\ \hfill & \quad -\frac{{m}^{2}}{4}\left[{G}_{;{p}^{\prime }}{G}_{;}^{{\enspace p}^{\prime }}+{G}_{;p}{G}_{;}^{\enspace p}+{m}^{2}{G}^{2}\right]\hfill \end{align} \tag{ 31 }$$

$$\begin{align}\hfill \text{Cov}[p(x){\epsilon}({x}^{\prime })]& =-\frac{({\partial }^{e}\phi {\partial }_{e}\phi )}{{\partial }_{{c}^{\prime }}\phi {\partial }_{{d}^{\prime }}\phi }{{G}_{;{c}^{\prime }}}_{p}{{G}_{;{d}^{\prime }}}^{p}\hfill \\ \hfill & \quad -\frac{1}{4}[{{G}_{;{p}^{\prime }}}_{q}{{G}_{;}}^{{p}^{\prime }q}+{m}^{2}\left(\right.{G}_{;{p}^{\prime }}{G}_{;}^{{\enspace p}^{\prime }}+{G}_{;p}{G}_{;}^{\enspace p}+{m}^{2}G]\hfill \end{align} \tag{ 32 }$$

$$\begin{align}\hfill \text{Cov}[{\epsilon}(x){\epsilon}({x}^{\prime })]& =\frac{1}{2}\frac{{({\partial }_{e}\phi {\partial }^{e}\phi )}^{2}}{({\partial }_{a}\phi {\partial }_{b}\phi {\partial }_{{c}^{\prime }}\phi {\partial }_{{d}^{\prime }}\phi )}({{G}_{;{c}^{\prime }}}_{b}{{G}_{;{d}^{\prime }}}_{a}+{{G}_{;{c}^{\prime }}}_{a}{{G}_{;{d}^{\prime }}}_{b})\hfill \\ \hfill & \quad -\frac{1}{2}\partial ({\partial }_{e}\phi {\partial }^{e}\phi )({\partial }_{a}\phi {\partial }_{b}\phi )({{G}_{;{p}^{\prime }}}_{b}{{G}_{{;}^{{p}^{\prime }}}}_{a}+{m}^{2}{G}_{;a}{G}_{;b})\hfill \\ \hfill & \quad +\frac{({\partial }_{e}\phi {\partial }^{e}\phi )}{({\partial }_{{c}^{\prime }}\phi {\partial }_{{d}^{\prime }}\phi )}{G}_{;{{c}^{\prime }}_{p}}{G}_{:{{d}^{\prime }}^{p}}+\frac{1}{4}\left\{{G}_{;{{p}^{\prime }}_{q}}{G}_{{;}^{{p}^{\prime }q}}\right.\hfill \\ \hfill & \quad \left.+{m}^{2}({G}_{;{p}^{\prime }}{G}_{;}^{{\enspace p}^{\prime }}+{G}_{;p}{G}_{;}^{\enspace p}+{m}^{2}G)\right\}\hfill \end{align} \tag{ 33 }$$

Thus we see that, there is one to one correspondence in the ideal fluid approximation of scalar field fluctuations. For the non-ideal case, as seen from equation (25) the two point covariances of energy density, heat flux and anisotropic stresses appear together in one expression. Thus it is not possible to get separate relations for these using the the present approach. This is an operational level difficulty, given the complexity of the equations and that the number of equations is less than the number of fluid variables in the non-ideal case. However this glitch shows up while trying to find out reverse relations only and not for the forward expressions that we get. In the following section we discuss the usefulness of our work.

The results in the above section are obtained by assuming negligible randomness and fluctuations for the four-velocity in the effective fluid approximation for scalar fields. This shows that stochastic fluctuations in the decoherence limit of quantum fields can be encapsulated in terms of generalized randomness of the bulk fluid variables, namely pressure, energy density etc in the hydro approximation.

We explain below in more detail, the possible cause for the physically validity of such non-thermal fluctuations in the system.

As stated in an earlier section, four-velocity of the effective fluid is a normalized quantity, hence the only possibility that it carries a random nature would be due to the corresponding unit vector fluctuating in various directions for different realizations of the field. This would happen if the scalar field retains all its quantum mechanical properties and coherence with phase and amplitude. Then the four-velocity fluctuations can approximately be associated with the fluid particles performing random motion similar to thermal effects as in Brownian motion scenario. As we have stated above, our results are obtained in the decoherence limit. The issue of decoherence of states is yet an open area of research, however it is known that we do not attain absolute classical results in the sense that the uncertainty due to Heisenberg's principle does not go exactly to zero in the decoherence limit. There are very recent advances which have been of observational consequences about foundations in quantum mechanics [8], which clearly show that for squeezed states it can be the magnitude of the scalar field responsible for such 'kicks' which can be observed on macroscopic objects, particularly in terms of radiation pressure. Though the origin of these kicks lies in the quantum noise, one can see the effect it has on macroscopic bodies, using radiation pressure in lasers. Similarly, but in a very different physical set up, the scalar field in an effective fluid approximation may show few common features in the decoherence limit in terms of a sub-hydro mesoscopic noise. This mesoscopic scale that we can touch upon in the decoherence limit in terms of the fluid bulk variables, has a new and very different characteristic in the description of dense matter. These fluctuations in the bulk physical quantities of the fluid can be linked to the fluctuations in magnitude of the scalar field. Such fluctuations can in fact be associated with a superfluid state of gravitating matter in relativistic stars. Thus the generalized fluctuations and randomness in the bulk quantities can show up as partial remnants of the fully quantum nature of the fields retained at these new mesoscopic scales after decoherence. How are these quantum fluctuations captured in bulk macroscopic quantities or are filtered through in the way we have shown here, is a deeper question, which needs further detailed research. The indication that we give here is towards the quantum potential and its possible fluctuations being responsible for the fluctuations of bulk quantities in the hydrodynamic limit. The quantum potential is not a kinetic quantity for scalar fields, and is essentially due to the inherent nature of the fields. For exotic matter in bulk under the influence of gravity, against which it supports itself from collapsing, this has several roles to play. For scalar field models of dark matter [19] (and references therein), it is the quantum potential that gives rise to the pressure in the fluid approximation. This is one of the directions which we would further like to explore in all details in upcoming work.

Our results indicate that, fluctuations of quantum fields can induce sub-hydro mesoscopic effects in the fluid description of matter. These can be given by 'generalized covariances' (or variances) of pressure, energy density, heat flux etc in the background spacetime. We give the first theoretical results (in continuation with [5]) in closed analytical form regarding fluctuation of matter fields as correspondences in the fluid approximation. In addition to being applicable for the perturbative theory in general relativity as the noise source, these fluctuations characterize the yet unexplored mesoscopic regime effects in the matter fields coupled to a spacetime of interest. The significance therefore, lies in realising their importance for compact astrophysical objects which are coupled to (say) thermal (or non-thermal) fields as discussed in [15, 16] and are of interest to collapsing clouds, towards critical phases and end states of collapse.

Thus our results can be used to analyse properties of dense compact matter in strong gravity regions at intermediate length scales which can be investigated further. An extended structure and properties given in terms of two point statistical covariances of matter fields being filtered out from microscales to effective classical variables in a mesoscopic description is the key feature in this article. In the currently developing area of semiclassical stochastic gravity, solutions of the semiclassical Einstein–Langevin equations have an intricate nature due to the presence of quantum stress tensor and its fluctuations, few results have been worked out [20–23] with all the rigour. With the version of matter fields in the hydrodynamic limit that we give, it is possible to find solutions applicable to the epoch after decoherence of the inflaton field sets in during the evolution of the Universe. Similarly for compact astrophysical systems one can find applications of this new theoretical formulation, as described earlier. Characterizing such a generalized stochastic source in the system is a first step towards this endeavour.

The usefulness of this correspondence can also be seen to give a direction for studying microscopic structure and its connections with kinetic theory in curved spacetime [13]. One can begin such an endeavour by trying to consider generalized fluctuations of matter fields instead of trying to define particles in a curved spacetime and construct the theory accordingly. We know that a global definition to particles and to vacuum in a curved spacetime background is not unique. One may then attempt to formulate a kinetic theory using the field fluctuations and its generalization as the basic entity. It is our future endeavour to investigate and explore on these lines of thought. With the framework of two point or higher correlations of fluctuations of matter fields, a tool to study non-local and extended structure of matter in the curved spacetime is brought into focus. Thus a kinetic theory of matter in curved spacetime can be based on these fluctuations rather than on the ambiguous localized particles. For such an aim, we would have to consider the full set of fluctuations including the kinetic term, without taking the decoherence limit. This would need the effective four-velocity vector and its fluctuations too as non-vanishing, giving rise to more terms in equation (17) and the set (18)–(21). We plan to carry this out in upcoming work.

The above plans for future call for more elaborate work and confirmation, further attempts of investigation and study in this regard are on the way.

The author is thankful to Nils Andersson for useful discussions and helpful suggestions. A major part of this work was carried out for the project funded by Department of Science and Technology (DST), India through Grant No. DST/WoS-A/2016/PM/100, for which the host institute IISER Pune, Department of Physical Sciences provided all facilities in the duration.

No new data were created or analysed in this study.