# The classical-quantum hybrid canonical dynamics and its difficulties with
special and general relativity

Lajos Diósi<EMAIL_ADDRESS>www.wigner.hu/~diosi Wigner Research Center
for Physics, H-1525 Budapest 114 , P.O.Box 49, Hungary  Eötvös Loránd
University, H-1117 Budapest, Pázmány Péter stny. 1/A

###### Abstract

We discuss the Hamiltonian hybrid coupling between a classical and a quantum
subsystem. If applicable to classical gravity coupled to quantized matter,
this hybrid theory might realize a captivating ‘postquantum’ alternative to
full quantum-gravity. We summarize the nonrelativistic hybrid dynamics in
improved formalism adequate to Hamiltonian systems. The mandatory decoherence
and diffusion terms become divergent in special and general relativistic
extensions. It is not yet known if any renormalization method might reconcile
Markovian decoherence and diffusion with relativity. Postquantum gravity could
previously only be realized in the Newtonian approximation. We argue that
pending problems of the recently proposed general relativistic postquantum
theory will not be solved if Markovian diffusion/decoherence are truly
incompatible with relativity.

##  I Introduction

The dynamical coupling between a classical and a quantum subsystem is of
multiple interests, e.g., in mathematical physics, in heuristic models, and
particularly in foundations. If gravity were fundamentally classical then its
hybridized dynamics with quantized matter would replace the mean-field
(semiclassical) approximation Møller (1962); Rosenfeld (1963) and the famously
inconclusive versions of quantum-gravity. Such captivating idea has been kept
alive from an episodic suggestion Maddox (1995); Anderson (1995) —based on
incorrect nonrelativistic (NR) hybrid dynamics Diósi (1996)— through works by
the present author and by others Diósi (1989, 1990, 1995); Diósi and Halliwell
(1998); Diósi (2011); Tilloy and Diósi (2016, 2017) until to culminate in
_postquantum_ gravity of Oppenheim and co-workers Oppenheim and Weller-Davies
(2022); Layton  _et al._ (2022); Oppenheim  _et al._ (2023a); Oppenheim
(2023).

Parallel to the fundamental concept, the underlying mathematical tool has been
researched persistently along important milestones Aleksandrov (1981);
Gerasimenko (1982); Boucher and Traschen (1988); Diósi (1995); Diósi and
Halliwell (1998); Diósi (2000); Diósi  _et al._ (2000); Diósi (2011, 2014);
Blanchard and Jadczyk (1993, 1995); Alicki and Kryszewski (2003); Oppenheim
_et al._ (2022, 2023b); Diósi (2023a). The central technical issue that has
been solved non-relativistically is the following. Suppose the hybrid
Hamiltonian containing in turn the classical Hamilton function of the
classical subsystem, the Hamilton operator of the quantum subsystem, and the
coupling between them:

$\hat{H}(q,p)=H_{\mathrm{C}}(q,p)+\hat{H}_{\mathrm{Q}}+\hat{H}_{\mathrm{CQ}}(q,p).$
(1)

The evolution equation of the state-vector of the quantum subsystem is the
Schrödinger equation $i\hbar|\Psi\rangle/dt=\hat{H}(q,p)|\Psi\rangle$. The
_backaction_ of the quantum subsystem on the classical one is non-trivial.
Towards the solution of interest, we introduce the hybrid state, represented
by the hybrid density $\hat{\rho}(q,p)\geq 0$ which is combination of the
density operator $\hat{\rho}_{\mathrm{Q}}=\int\hat{\rho}(q,p)dqdp$ of the
quantum subystem and the phase space density
$\rho_{\mathrm{C}}(q,p)=\mathsf{tr}\hat{\rho}(q,p)$ of the classical one.
Assume the following combination of the classical and quantum dynamics
Aleksandrov (1981):

$\displaystyle\frac{d\hat{\rho}(q,p)}{dt}$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}[\hat{H}(q,p),\hat{\rho}]+\mathbb{H}\\{\hat{H}(q,p),\hat{\rho}(q,p)\\}$
$\displaystyle\equiv$
$\displaystyle\\{\hat{H}(q,p),\hat{\rho}(q,p)\\}_{\mathrm{A}}$

where $\\{~{},~{}\\}$ stands for the Poisson bracket. The term
$\mathbb{H}\\{\hat{H}_{CQ}(q,p),\hat{\rho}(q,p)\\}$ represents the backaction,
the symbol $\mathbb{H}$ means the Hermitian part. If it is zero, we get the
standard classical and quantum dynamics separately for the two subsystems, as
we should. But the seemingly plausible dynamics (I) is not yet mathematically
correct, it does not preserve the positivity of $\hat{\rho}(q,p)$. Additional
decoherence and diffusion mechanisms are mandatory and they are subject of
trade-off: stronger decoherence allows for weaker diffusion and vice versa
Diósi (1995). The ultimate general form of hybrid NR dynamics appeared in
refs. Oppenheim  _et al._ (2022, 2023b); Diósi (2023a, b).

Instead of a master equation for $\hat{\rho}(q,p)$, stochastic differential
equations for the pure quantum state $\hat{P}$ and the classical variables
$(q,p)$ offer an equivalent alternative. As an analogy, remember for example
that the classical Fokker–Planck equation is equivalent with the Langevin
stochastic differential equation. In the hybrid case, the backaction is
realized by time-continuous quantum measurement — _monitoring_ — of the
quantum subsystem and _feedback_ of the measured signal into the classical
subsystem. The importance of this formalism is emphasized especially in refs.
Diósi (2023a); Tilloy (2018). Compared to the master equation of hybrid
canonical coupling, the modular monitoring-plus-feedback construction gives
better intuition as observed in ref. Tilloy (2018).

Our goal is threefold: a convenient introduction to the mathematics of NR
hybrid canonical dynamics, the assessment of its application in postquantum
gravity and a discussion if it could have surpassed its old Newtonian
‘forerunner’.

Section II recapitulates state-of-the-art knowledge of NR hybrid canonical
dynamics. Section III explains the locality condition of relativistic
invariance and the resulting divergences. Section IV tests the special
relativistic extension on the simplest example of hybrid coupling between a
classical and a quantized scalar field. Section V revisits the effort towards
general relativistic postquantum gravity, extending the NR hybrid dynamics for
general relativity. Section VI recapitulates the NR ‘forerunner’ of
postquantum general relativity. Final remarks and our conclusion are given by
sec. VII.

##  II The nonrelativistic canonical hybrid dynamics

Our hybrid system of interest consists of a NR classical canonical subsystem
and a NR quantized subsystem. To model their coupled dynamics we start from
the naive combination (I). In addition to the Dirac and Poisson brackets there
are mandatory decoherence and diffusion terms which will necessitate the
postulation of a Riemann metric on the phase space manifold (or on its
submanyfold). The resulting irreversible dynamics obtains the form of the
hybrid master equation (HME) which is the combination of the classical
Fokker–Planck and the quantum Lindblad equations (sec. II.1). This
irreversible dynamics is equivalent with the coupled stochastic processes in
the classical phase space and the Hilbert space, respectively, and represented
by a couple of hybrid stochastic differential equations (HSDEs) in sec. II.2.
In physics, the special case is of interest, when the classical coordinates
are coupled to the quantum subsystem but the classical momenta aren’t (sec.
II.3). The material presented here is based primarily on refs. Oppenheim  _et
al._ (2022, 2023b); Diósi (2023a, b), and deduced basically from Diósi (2023a)
(cf. Appendix A), improved by the Riemann metric interpretation of the
decoherence and diffusion kernels. It is important that we treat the HME and
HSDE formalisms as equivalent, both have their own conceptual universality.

###  II.1 Hybrid master equation

Let $\hat{H}(x)\equiv\hat{H}(q,p)$ be a our hybrid Hamiltonian where the
classical subsystem is canonical. The first $N$ canonical variables
$\\{x^{n};n=1,\dots,N\\}$ are the coordinates and the second $N$ ones
$\\{x^{n};n=N+1,\dots,2N\\}$ are the momenta:

$x^{n}=\left\\{\begin{array}[]{cl}q_{n};&n=1,2,\dots N\\\
p^{n};&n=N+1,N+2,\dots,2N.\end{array}\right.$ (3)

The HME of the hybrid density $\hat{\rho}(q,p)=\hat{\rho}(x)$ takes this form:

$\frac{d\hat{\rho}}{dt}=-\frac{i}{\hbar}[\hat{H},\hat{\rho}]+\mathbb{H}\\{\hat{H},\hat{\rho}\\}+\mathcal{D}\hat{\rho}\equiv\\{\hat{H},\hat{\rho}\\}_{\mathrm{A}}+\mathcal{D}\hat{\rho},$
(4)

where $\\{~{},~{}\\}_{\mathrm{A}}$ is the Aleksandrov hybrid bracket,
$\mathcal{D}$ is the superoperator of decoherence and diffusion (D&D). The
notation of $x$-dependences of $\hat{H},\hat{\rho},\mathcal{D}$ are spared.
The classical canonical Poisson bracket is defined by

$\\{\hat{A},\hat{B}\\}=\hat{A}_{,n}\epsilon^{nm}\hat{B}_{,m}=(\hat{A}\epsilon^{nm}\hat{B}_{,m})_{,n}=(\hat{A}_{,n}\epsilon^{nm}\hat{B})_{,m},$
(5)

where $\epsilon^{nm}$ is the $2N\times 2N$ symplectic matrix. We introduced
the shorthand notation for partial derivatives like $\partial\hat{A}/\partial
x^{n}=\hat{A}_{,n}$ as well as Einstein convention for summation of repeated
indices. If we define the canonical velocity operators

$\hat{v}^{n}=\\{x^{n},\hat{H}\\}=-\epsilon^{nm}\hat{H}_{,m}$ (6)

then the Poisson bracket will have the useful equivalent expression:

$\mathbb{H}\\{\hat{H},\hat{\rho}\\}=\mathbb{H}\hat{v}^{n}\hat{\rho}_{,n}.$ (7)

To construct the canonical HME we impose a Riemann metric structure in
addition to the symplectic structure of the phase space, via the arbitrary
choice of the $2N\times 2N$ covariant metric tensor $\gamma_{nm}(x)$. The D&D
terms are the following:

$\displaystyle\mathcal{D}\hat{\rho}$ $\displaystyle=$
$\displaystyle-\frac{\gamma_{nm}}{8}[\hat{v}^{n},[\hat{v}^{m},\hat{\rho}]]+\tfrac{1}{2}\left(\gamma^{nm}\hat{\rho}\right)_{,nm}$
(8) $\displaystyle\equiv$
$\displaystyle\mathcal{D}_{\mathrm{Q}}\hat{\rho}+\mathcal{D}_{\mathrm{C}}\hat{\rho},$

where we assume that the velocities $\hat{v}^{n}(x)$ are linearly independent
operators, also independent from any c-number functions. That is, we assume
the equation

$\lambda_{n}(x)\hat{v}^{n}(x)=\varphi(x)$ (9)

is satisfied only for vanishing $\lambda_{n}$ and $\varphi$.

###  II.2 Hybrid stochastic differential equations

The canonical HME (4) with D&D (8) is equivalent with two coupled stochastic
processes, one for the diffusion of the pure state
$\hat{P}_{t}\equiv|\Psi_{t}\rangle\langle\Psi_{t}|$ in the Hilbert space, the
other one for the diffusion of $x_{t}$ in the phase space, meaning in fact the
statistical interpretation of the HME. Also called stochastic _unraveling_ of
the HME, the processes are defined by the couple of HSDEs:

$\displaystyle\\!\\!\\!\\!\\!\\!\frac{d\hat{P}}{dt}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!-\frac{i}{\hbar}[\hat{H}(\\!x\\!),\\!\hat{P}]\\!+\\!\mathcal{D}_{\mathrm{Q}}(\\!x\\!)\hat{P}\\!+\\!\mathbb{H}(\hat{v}^{n}(\\!x\\!)\\!-\\!\langle\hat{v}^{n}(\\!x\\!)\rangle)\hat{P}w_{n}(\\!x\\!)$
$\displaystyle\\!\\!\\!\frac{dx^{n}}{dt}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\langle\hat{v}^{n}(x)\rangle+w^{n}(x)$ (11)

where $\langle\hat{v}^{n}(x)\rangle=\mathsf{tr}(\hat{v}^{n}(x)\hat{P})$. Both
SDEs are driven by the same white-noise $w_{n}=\gamma_{nm}w^{m}$ whose
correlations are determined by the metric:

$\displaystyle\mathbb{M}w^{n}(x,t)w^{m}(x,\tau)$ $\displaystyle=$
$\displaystyle\gamma^{nm}(x)\delta(t-\tau)$
$\displaystyle\mathbb{M}w_{n}(x,t)w_{m}(x,\tau)$ $\displaystyle=$
$\displaystyle\gamma_{nm}(x)\delta(t-\tau)$
$\displaystyle\mathbb{M}w^{n}(x,t)w_{m}(x,\tau)$ $\displaystyle=$
$\displaystyle\delta^{n}_{m}\delta(t-\tau).$ (12)

The symbol $\mathbb{M}$ stands for the stochastic mean.

In this formalism of the hybrid dynamics the backaction follows from the
monitoring-plus-feedback mechanism. The eq. (LABEL:dP) coincides with the
stochastic master equation of time-continuous simultaneous quantum
measurements —monitoring— of the observables $\hat{v}^{n}$. The measured
signal $\langle\hat{v}^{n}\rangle+w^{n}$ will then control a feedback in the
equation of motion (11) of the classical phase space variables $x^{n}$. Note
that this SDE can be written as

$\frac{dx^{n}}{dt}=\\{x^{n},\langle\hat{H}(x)\rangle\\}+w^{n},$ (13)

which is the mean-field (semiclassical) backaction plus our mandatory white-
noise. Observe that unlike white-noises, the phase-space coordinates
$x^{n}(t)$ are continuous functions, containing the integrals of the white-
noises $w^{n}(t)$. The path in phase space is a (generalized) Wiener-process.

###  II.3 Coordinate coupling

The D&D terms (8) correspond to the minimum noise dynamics if the $2N$
velocities $\hat{v}^{n}(x)$ are independent operator fields on the phase
space. However, they are not so in many concrete hybrid systems. Suppose $K$
is the maximum number of independent constraints (9):

$\lambda^{a}_{n}(x)\hat{v}^{n}(x)=\varphi^{a}(x),~{}~{}~{}(a=1,2,\dots,K)$
(14)

with $K$ linear independent vector fields $\lambda^{a}_{n}\neq 0$. Then we can
always find a coordinate transformation $x^{n}\Rightarrow f^{n}(x)$ such that
the first $2N-K$ velocities $\hat{v}^{n}$ become independent operators and the
rest of them are c-numbers: $\hat{v}^{n}=v^{n}\hat{I}$ for $n\rangle 2N-K$.
Then the minimum noise D&D corresponds to the same structure (8) but the
indices run from $1$ to $2N-K$. The $(2N-K)\times(2N-K)$ metric tensor
$\gamma_{nm}$ defines a Riemann structure on the first $2N-K$ coordinates
while it depends parametrically on the rest of them.

An important special case is coordinate-coupling when
$\partial\hat{H}/\partial q^{n}$ are independent operators but
$\partial\hat{H}/\partial p^{n}$ are zeros or c-number functions. We impose
the Riemann metric structure on the subspace of canonical coordinates only.
The $N\times N$ metric tensor $\gamma_{nm}(q,p)$ will be the metric for the
coordinates $q$ still it may parametrically depend on the momenta $p$ as well.
With momentum velocity operators

$\hat{v}^{n}=-\frac{\partial\hat{H}}{\partial
q_{n}}=-\frac{\partial\hat{H}_{\mathrm{CQ}}}{\partial q_{n}},$ (15)

the D&D terms take this form:

$\mathcal{D}\hat{\rho}=-\frac{\gamma_{nm}}{8}\left[\frac{\partial\hat{H}_{\mathrm{CQ}}}{\partial
q_{n}},\left[\frac{\partial\hat{H}_{\mathrm{CQ}}}{\partial
q_{m}},\hat{\rho}\right]\right]+\frac{1}{2}\frac{\partial^{2}\left(\gamma^{nm}\hat{\rho}\right)}{\partial
p^{n}\partial p^{m}}~{}.$ (16)

As we see, momentum velocity operators $\hat{v}^{n}$ are actors of decoherence
classical momenta $p^{n}$ are subjects of diffusion.

The HSDEs (11,LABEL:dP) of the equivalent stochastic processes become the
following:

$\displaystyle\frac{d\hat{P}}{dt}$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}[\hat{H}(q,p),\hat{P}]+\mathcal{D}(q,p)\hat{P}+$
(17)
$\displaystyle+\mathbb{H}\left(\hat{v}^{n}(q,p)-\langle\hat{v}^{n}(q,p)\rangle\right)\hat{P}w_{n}(q,p)$
$\displaystyle\frac{dq_{n}}{dt}$ $\displaystyle=$
$\displaystyle\frac{\partial\langle\hat{H}(q,p)\rangle}{\partial p^{n}}$ (18)
$\displaystyle\frac{dp^{n}}{dt}$ $\displaystyle=$
$\displaystyle-\frac{\partial\langle\hat{H}(q,p)\rangle}{\partial
q_{n}}+w^{n}(q,p).$ (19)

Like in eq. (II.2), the noise $w^{n}=\gamma^{nm}w_{m}$ satisfies

$\displaystyle\mathbb{M}w^{n}(q,p,t)w^{m}(q,p,\tau)$ $\displaystyle=$
$\displaystyle\gamma^{nm}(q,p)\delta(t-\tau)$
$\displaystyle\mathbb{M}w_{n}(q,p,t)w_{m}(q,p,\tau)$ $\displaystyle=$
$\displaystyle\gamma_{nm}(q,p)\delta(t-\tau)$
$\displaystyle\mathbb{M}w^{n}(q,p,t)w_{m}(q,p,\tau)$ $\displaystyle=$
$\displaystyle\delta^{n}_{m}\delta(t-\tau).$ (20)

This is the minimum-noise D&D term of general coordinate coupling provided the
derivatives $\partial\hat{H}/\partial q_{n}$ are $N$ independent operators.

##  III Locality condition of relativistic continuum dynamics

Let us consider the Markovian dynamics $d\rho/dt=L\rho$ where $\rho$ is
classical, quantum, or hybrid state and $L$ is the generator of time evolution
respectively of Fokker–Planck, Lindblad, or hybrid field dynamics. For
relativistic invariance, $L$ must be the zeroth component of a four-vector.
This condition on $L$ is, however, not sufficient Diósi (2022). It must be the
spatial integral of the generator density $\mathcal{L}(\mathbf{r})$:

$L=\int\mathcal{L}(\mathbf{r})d\mathbf{r}$ (21)

and $\mathcal{L}(\mathbf{r})$ must satisfy the locality condition

$[\mathcal{L}(\mathbf{r}),\mathcal{L}(\mathbf{s})]=0.$ (22)

Then, given the state on the hypersurface $\sigma_{1}$, it maps to another
hypersurface as follows:

$\rho(\sigma_{2})=\exp\left(\int_{\sigma_{2}\succ(t,\mathbf{r})\succ\sigma_{1}}\hskip-40.0pt\mathcal{L}(\mathbf{r})d\mathbf{r}dt\right)\rho(\sigma_{1}).$
(23)

Without the locality condition this relationship does not exist and we miss
the map between states on two different hipersurfaces. Of course the map
between Lorentz frames is also impossible.

In standard relativistic field theories, classical or quantum, the generator
field reads $\mathcal{L}=\\{\mathcal{H},~{}\\}$ or
$\mathcal{L}=-(i/\hbar)[\hat{\mathcal{H}},~{}]$, respectively, and is local
since the Hamiltonian densities $\mathcal{H},\hat{\mathcal{H}}$ are local.
Locality of the generator $\mathcal{L}$ survives in effective field theories.
If, however, the effective theory contains diffusion (or decoherence) then we
face difficulties. To retain locality of the generator $\mathcal{L}$ the
diffusion (decoherence) kernel must be local, i.e., proportional to
$\delta(\mathbf{r}-\mathbf{s})$ and then, unfortunately, the theory yields
infinities. Take, for instance, the Fokker–Planck equation of a scalar field
with the local diffusion kernel $\gamma\delta(\mathbf{r}-\mathbf{s})$. It
yields infinite rate kinetic energy production at each point $\mathbf{r}$. It
is not known whether relativistic Fokker–Planck field equations are
renormalizable or aren’t. The same concern applies to the Lindblad and hybrid
dynamics.

##  IV On special relativistic hybrid field dynamics

We test the NR hybrid classical-quantum theory (sec. II) in coordinate
coupling (sec. II.3) of special relativistic fields. The coordinates and
momenta become functions $q(\mathbf{r}),p(\mathbf{r})$, the discrete labels
$n,m$ become the continuous spatial vectors $\mathbf{r},\mathbf{s}$,
respectively. Sums over indices become spatial integrals, Kronecker deltas
become Dirac deltas, derivations like e.g. $\partial/\partial q_{n}$ become
functional derivations $\delta/\delta q(\mathbf{r})$.

Consider the coupling of the free classical scalar field $q(\mathbf{r})$ (with
canonical momentum $p(\mathbf{r})$) to the free quantized boson field
$\hat{\phi}(\mathbf{r})$ (with canonical momentum $\hat{\pi}(\mathbf{r})$):

$\hat{H}_{\mathrm{CQ}}[q]=\kappa\int
q(\mathbf{r})\hat{\phi}(\mathbf{r})d\mathbf{r}.$ (24)

This coupling is independent of the classical canonical momentum
$p(\mathbf{r})$ and we can apply the eq. (16) with

$\frac{\delta\hat{H}_{\mathrm{CQ}}}{\delta
q(\mathbf{r})}=-\kappa\hat{\phi}(\mathbf{r}).$ (25)

The D&D terms depend on the metric which can in general be a functional kernel
$\gamma_{[q,q^{\prime}]}$. At the same time, we should damp remote correlation
in decoherence as well as in diffusion. The metric must have a spatial damping
factor. In the simplest case, we choose a flat metric
$\gamma_{\mathbf{r}\mathbf{r}^{\prime}}$ without the functional dependences.
The covariant and contravariant kernels are inverses of each other:

$\int\gamma_{\mathbf{r}\mathbf{s}^{\prime}}\gamma^{\mathbf{s}^{\prime}\mathbf{s}}d\mathbf{r}^{\prime}=\delta(\mathbf{r}-\mathbf{s}).$
(26)

Then the D&D terms (16) take the following form:

$\displaystyle\mathcal{D}\hat{\rho}=$ $\displaystyle-$
$\displaystyle\frac{\kappa^{2}}{8}\int\int\gamma_{\mathbf{r}\mathbf{s}}\left[\hat{\phi}(\mathbf{r}),\left[\hat{\phi}(\mathbf{s}),\hat{\rho}\right]\right]d\mathbf{r}d\mathbf{s}$
(27) $\displaystyle+$
$\displaystyle\frac{1}{2}\int\int\gamma^{\mathbf{r}\mathbf{s}}\frac{\delta^{2}\left(\hat{\rho}\right)}{\delta
p(\mathbf{r})\delta p(\mathbf{s})}d\mathbf{r}d\mathbf{s}.$

Both D&D violate the special relativistic invariance unless the kernel itself
is invariant. It is easy to ensure Galilean invariance if
$\gamma_{\mathbf{r}\mathbf{s}}$ is function of $|\mathbf{r}-\mathbf{s}|$. The
only kernels that ensure relativistic invariance are the singular local ones:

$\gamma_{\mathbf{r}\mathbf{s}}=\gamma\delta(\mathbf{r}-\mathbf{s}),~{}~{}~{}~{}\gamma^{\mathbf{r}\mathbf{s}}=\gamma^{-1}\delta(\mathbf{r}-\mathbf{s}).$
(28)

But they lead to untractable divergences of the kinetic energy density
$\mathcal{K}=\tfrac{1}{2}(\hat{\pi}^{2}+p^{2})$:

$\frac{d\mathcal{K}(\mathbf{r})}{dt}=\frac{1}{2}\mathcal{D}_{\mathrm{Q}}^{\dagger}\pi^{2}(\mathbf{r})+\frac{1}{2}\mathcal{D}_{\mathrm{C}}^{\dagger}p^{2}(\mathbf{r})=\left(\frac{\gamma}{4\hbar^{2}}+\frac{1}{\gamma}\right)\delta(\mathbf{0}).$
(29)

The D&D terms (27) yield infinite rate of heating at each location in the
quantized bosonic as well as in the classical scalar field subsystems.
Allowing functional dependence of the metric does not help since the
relativistic invariance of spatial damping requires the presence of the
spatial $\delta$ function singularity.

These divergences are different from the usual divergences in relativistic
field theory. Either we invent their renormalization, if it is possible at
all, or we are loosing special relativistic invariance, and we are left with
the NR hybrid calculus.

##  V On hybrid general relativity

Instead of full quantum-gravity, it were of great simplification if we could
keep the space-time classical. Accordingly, we take a chance to extend the NR
hybrid dynamics of Sec. II for coupling between classical canonical form of
general relativity and quantized relativistic matter. In canonical form of
Einstein’s general relativity, $(3+1)$-dimensional diffeomorphism invariance
is encoded by the combination of $3$-dimensional spatial diffeomorphism (sDM)
invariance and time-reparametrization (tRP) invariance. Following refs.
Oppenheim  _et al._ (2023a); Oppenheim (2023), we build up the formal sDM and
tRP invariant hybrid equations (sec. V.1). We are _going to the wall_ to
ensure both these ivariances but that remains a problem ( V.2).

###  V.1 Equivalent formalisms: HME and HSDE

The canonical coordinates are the configurations of the $3\times 3$ metric
tensor field $g_{ik}(\mathbf{r})$, satisfying the canonical commutation
relationship with the canonical momenta $\pi^{ik}(\mathbf{r})$:

$\\{g_{ij}(\mathbf{r}),\pi^{kl}(\mathbf{s})\\}=\delta_{ij}^{kl}\delta(\mathbf{r},\mathbf{s}),$
(30)

where
$\delta_{ij}^{kl}=\frac{1}{2}(\delta_{i}^{k}\delta_{j}^{l}+\delta_{i}^{l}\delta_{j}^{k})$
and we use the covariant delta function

$\delta(\mathbf{r},\mathbf{s})=\frac{1}{\sqrt{g(\mathbf{r})}}\delta(\mathbf{r}-\mathbf{s})~{}.$
(31)

where $g=\mathrm{det}d_{ij}$. The covariant Poisson bracket is defined by

$\\{\hat{A},\hat{B}\\}=\int\left(\frac{\delta\hat{A}}{\delta
g_{ij}(\mathbf{r})}\frac{\delta\hat{B}}{\delta\pi^{ij}(\mathbf{r})}-\frac{\delta\hat{A}}{\delta\pi^{ij}(\mathbf{r})}\frac{\delta\hat{B}}{\delta
g_{ij}(\mathbf{r})}\right)dV,$ (32)

where $dV=dV_{\mathbf{r}}=\sqrt{g(\mathbf{r})}d\mathbf{r}$. Through this
section, the functional derivatives are the covariant ones, i.e., $1/\sqrt{g}$
times the common ones.

The hybrid Hamiltonian reads:

$\hat{H}[g,\pi;N,\vec{N}]=H_{\mathrm{G}}[g,\pi;N,\vec{N}]+\hat{H}_{\mathrm{M}}[g;N],$
(33)

where $H_{\mathrm{G}}[g,\pi;N,\vec{N}]$ is the classical Hamilton function of
gravity and $\hat{H}_{\mathrm{M}}[g;N]$ is the Hamiltonian of the quantized
matter fields, coupled only to $g_{ik}$ and not to $\pi^{ik}$. They depend on
the freely chosen lapse $N$ and shift $N_{i}$:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!H_{\mathrm{G}}[g,\pi;N,\vec{N}]\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\int\\!\\!\left(N(\mathbf{r})\mathcal{H}_{\mathrm{G}}(\mathbf{r})\\!+\\!N_{i}(\mathbf{r})\mathcal{P}^{i}_{G}(\mathbf{r})\right)\\!dV,$
(34) $\displaystyle\hat{H}_{\mathrm{M}}[g;N]\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\int\\!\\!\left(N(\mathbf{r})\hat{\mathcal{H}}_{\mathrm{M}}(\mathbf{r})+N_{i}\mathcal{P}^{i}_{M}(\mathbf{r})\right)dV.$
(35)

$\mathcal{H}_{\mathrm{G}}(\mathbf{r})$ and
$\hat{\mathcal{H}}_{\mathrm{M}}(\mathbf{r})$ are the Hamiltonian densities of
gravity and matter, respectively, and $\mathcal{P}_{G}^{i}$ is the momentum
density of gravity:

$\mathcal{P}^{i}_{G}=-2\nabla_{i}\pi^{ij}(\mathbf{r}),$ (36)

where $\nabla_{\\!j}$ denotes covariant derivation. The gravity’s Hamiltonian
density reads:

$\mathcal{H}_{G}=\frac{16\pi
G}{c^{2}}\frac{1}{g}\left(\pi^{ij}\pi_{ij}-\tfrac{1}{2}(\pi^{i}_{i})^{2}\right)-\frac{c^{4}}{16\pi
G}R,$ (37)

with the scalar curvature $R$. The matter’s Hamiltonian
$\hat{\mathcal{H}}_{M}(\mathbf{r})$ and momentum density $\mathcal{P}^{i}_{M}$
depend on the matter fields. Remember that they should not depend on
$\pi^{ik}$ .

In the hybrid Hamiltonian (33), the lapse $N$ multiplies the Hamiltonian
constraint, the shift $N_{i}$ multiplies the diffeomorphism constraint which
we impose on the hybrid state:

$\displaystyle\left(\mathcal{H}_{\mathrm{G}}(\mathbf{r})+\hat{\mathcal{H}}_{\mathrm{M}}(\mathbf{r})\right)\hat{\rho}[g,\pi]=0,$
(38)
$\displaystyle\left(\mathcal{P}^{i}_{G}(\mathbf{r})+\hat{\mathcal{P}}^{i}_{M}(\mathbf{r})\right)\hat{\rho}[g,\pi]=0.$
(39)

These might ensure tRP and sDM invariances respectively. The conditional
phrase is of reason. If both gravity and matter were quantized (or classical),
then the above constraints would guarantee the said invariances under pure
classical canonical (or pure unitary) dynamics. Their compatibility and
applicability in hybrid dynamics are not yet clear. Moreover, hybrid dynamics
are not necessarily compatible with tRP and sDM invariances, as we see below.

To construct the hybrid coupling and the D&D terms, we need the momentum
velocity operators (15):

$\hat{v}^{ik}(\mathbf{r})=-\frac{\delta\hat{H}_{M}}{\delta
g_{ik}(\mathbf{r})}=-N(\mathbf{r})\\!\left(\\!\frac{\partial\hat{\mathcal{H}}_{M}(\mathbf{r})}{\partial
g_{ik}(\mathbf{r})}+\frac{1}{2}g^{ik}(\mathbf{r})\hat{\mathcal{H}}_{M}(\mathbf{r})\\!\right).$
(40)

The HME (4) of the state $\hat{\rho}[g,\pi]$ takes this form:

$\frac{d\hat{\rho}}{dt}=-\frac{i}{\hbar}[\hat{H}_{M},\hat{\rho}]+\\{H_{G},\hat{\rho}\\}-\mathbb{H}\int\hat{v}^{ik}\frac{\delta\hat{\rho}}{\delta\pi^{ik}}dV+\mathcal{D}\hat{\rho}\\\
$ (41)

While the hybrid Hamiltonian parts are unique, the D&D term
$\mathcal{D}\hat{\rho}$ is not, its consistent choice is nontrivial, see sec.
V.2.

The HME (41) has its alternative stochastic representation in terms of SHDEs.
We apply the eqs. (18-II.3):

$\displaystyle\\!\\!\\!\frac{d\hat{P}}{dt}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!-\frac{i}{\hbar}[\hat{H}_{\mathrm{M}},\hat{P}]\\!+\\!\mathcal{D}_{\mathrm{Q}}\hat{P}\\!+\\!\\!\mathbb{H}\\!\\!\int\\!\\!\left(\hat{v}^{ij}\\!\\!-\\!\\!\langle\hat{v}^{ij}\rangle\right)\\!\\!\hat{P}w_{ij}dV$
(42) $\displaystyle\\!\\!\\!\frac{dg_{ij}}{dt}\\!$ $\displaystyle=$
$\displaystyle\frac{\delta H_{\mathrm{G}}}{\delta\pi^{ij}}$ (43)
$\displaystyle\\!\\!\\!\frac{d\pi^{ij}}{dt}\\!$ $\displaystyle=$
$\displaystyle-\frac{\delta H_{\mathrm{G}}}{\delta
g_{ij}}+\langle\hat{v}^{ij}\rangle+w^{ij}$ (44)

where the noises satisfy

$\displaystyle\mathbb{M}w^{ij}(\mathbf{r},t)w^{kl}(\mathbf{s},\tau)$
$\displaystyle=$
$\displaystyle\gamma^{ij|kl}(\mathbf{r}|\mathbf{s})\delta(t-\tau)$
$\displaystyle\mathbb{M}w_{ij}(\mathbf{r},t)w_{kl}(\mathbf{s},\tau)$
$\displaystyle=$
$\displaystyle\gamma_{ij|kl}(\mathbf{r}|\mathbf{s})\delta(t-\tau)$
$\displaystyle\mathbb{M}w^{ij}(\mathbf{r},t)w_{kl}(\mathbf{s},\tau)$
$\displaystyle=$ $\displaystyle\delta^{ij}_{kl}\delta(t-\tau),$ (45)

and $\mathcal{D}_{\mathrm{Q}}$ will be discussed in sec. V.2.

As we said in sec. II.2, the eq. (42) corresponds to the quantum monitoring of
the velocity operators $\hat{v}^{ij}=-\delta\hat{H}_{\mathrm{M}}/\delta
g_{ij}$ and the noisy measured signal $\langle\hat{v}^{ij}\rangle+w^{ij}$ is
fed back on the rhs of the eq. (44) of $d\pi^{ij}/dt$.

###  V.2 The decoherence-diffusion kernels

Recall that the hybrid dynamics (sec. II.3) assumes a certain metric on the
space of canonical coordinates, which is a functional metric on the function
space of $3\times 3$ metric tensor fields $g_{ij}(\mathbf{r})$. We restrict
ourselves for the metrics

$(dg)^{2}=\int\int\gamma^{ij|kl}(\mathbf{r}|\mathbf{s})dg_{ij}(\mathbf{r})dg_{kl}(\mathbf{s})dV_{\mathbf{r}}dV_{\mathbf{s}},$
(46)

where the functional metric tensor $\gamma$ contains explicit coordinate
dependence on $(\mathbf{r},\mathbf{s})$ to damp remote correlations, also it
may depend on $g_{ij}(\mathbf{r})$ and $g_{kl}(\mathbf{s})$ (meaning nonflat
functional geometry). Accordingly, the D&D terms take this form:

$\displaystyle\mathcal{D}_{\mathrm{Q}}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!-\frac{1}{8}\\!\int\\!\\!\\!\int\\!\\!\gamma^{-1}_{ij|kl}(\mathbf{r}|\mathbf{s})\left[\hat{v}^{ij}(\mathbf{r}),[\hat{v}^{kl}(\mathbf{s}),\hat{\rho}]\right]dV_{\mathbf{r}}dV_{\mathbf{s}}$
(47) $\displaystyle\mathcal{D}_{\mathrm{C}}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int\\!\\!\\!\int\\!\frac{\delta^{2}\left(\gamma^{ij|kl}(\mathbf{r}|\mathbf{s})\hat{\rho}\right)}{\delta\pi^{ij}(\mathbf{r})\delta\pi^{kl}(\mathbf{s})}dV_{\mathbf{r}}dV_{\mathbf{s}},$
(48)

where the covariant and contravariant metrics satisfy the functional
relationship

$\int\int\gamma_{ij|k^{\prime}l^{\prime}}(\mathbf{r}|\mathbf{s}^{\prime})\gamma^{k^{\prime}l^{\prime}|kl}(\mathbf{s}^{\prime}|\mathbf{s})dV_{\mathbf{s}^{\prime}}=\delta_{ij}^{kl}\delta(\mathbf{r},\mathbf{s})~{}.$
(49)

It is instructive to consider the simple special case when the kernels are
local. Then their structure is perfectly determined by covariance:

$\displaystyle\gamma_{ij|kl}(\mathbf{r}|\mathbf{s})$ $\displaystyle=$
$\displaystyle\frac{\gamma(R)}{N(\mathbf{r})}G^{(\alpha)}_{ij|kl}(\mathbf{r})\delta(\mathbf{r},\mathbf{s})$
(50) $\displaystyle G^{(\alpha)}_{ij|kl}$ $\displaystyle=$
$\displaystyle\tfrac{1}{2}g_{ik}g_{jl}+\tfrac{1}{2}g_{il}g_{jk}+\alpha
g_{ij}g_{kl}$ $\displaystyle\gamma^{ij|kl}(\mathbf{r}|\mathbf{s})$
$\displaystyle=$
$\displaystyle\frac{N(\mathbf{r})}{\gamma(R)}G_{(\beta)}^{ij|kl}(\mathbf{r})\delta(\mathbf{r},\mathbf{s})$
(51) $\displaystyle G_{(\beta)}^{ij|kl}$ $\displaystyle=$
$\displaystyle\tfrac{1}{2}g^{ik}g^{jl}+\tfrac{1}{2}g^{il}g^{jk}+\beta
g^{ij}g^{kl}.$

These kernels are positive if $\alpha,\beta\rangle-1/3$. If
$3\alpha\beta+\alpha+\beta=0$ then the kernels are each other’s inverses as
they should, according to eq. (49).

With the above kernels, unfortunately, both the D&D terms in eqs. (47) and
(48), resp., become divergent because of the $\delta$-functions, just like in
sec. IV. However, a rescue procedure seems to be on offer.

We could try the sDM invariant regularization. For instance, we replace the
$\delta(\mathbf{r},\mathbf{s})$ in the decoherence kernel (50) by

$\mathcal{N}_{\epsilon}(\mathbf{r},\mathbf{s})\exp\left(-\frac{\ell^{2}(\mathbf{r},\mathbf{s})}{2\epsilon}\right)$
(52)

where $\mathcal{N}_{\epsilon}(\mathbf{r},\mathbf{s})$ is for normalization,
$\ell(\mathbf{r},\mathbf{s})$ is the geodesic distance between $\mathbf{r}$
and $\mathbf{s}$, and $\epsilon$ is the small parameter to go to $+0$. To keep
covariance, the index factor, too, should go nonlocal:

$\displaystyle G^{(\alpha)}_{ij|kl}(\mathbf{r}|\mathbf{s})\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\tfrac{1}{2}P^{j^{\prime}}_{j}\bar{P}^{k^{\prime}}_{k}g_{ik^{\prime}}(\mathbf{r})g_{j^{\prime}l}(\mathbf{s})\\!+\\!\tfrac{1}{2}P^{j^{\prime}}_{j}\bar{P}^{l^{\prime}}_{l}g_{il^{\prime}}(\mathbf{r})g_{kj^{\prime}}(\mathbf{s})$
(53) $\displaystyle+\alpha g_{ij}(\mathbf{r})g_{kl}(\mathbf{s}).$

Here $P^{i}_{j}$ is geodesic parallel transport of covariant vectors from
$\mathbf{s}$ to $\mathbf{r}$ and $\bar{P}^{i}_{j}$ is the same from
$\mathbf{r}$ to $\mathbf{s}$.

So far so good. The problem is the factor $1/N(\mathbf{r})$ which ensures the
tRP invariance. We should keep it but we cannot. It can not be split for the
two locations $\mathbf{r}$ and $\mathbf{s}$. The same problem would come along
with the factor $N(\mathbf{r})$ if we regularized the decoherence kernel (51)
first.

The lesson goes beyond the example. Any nonlocal generalization of the kernels
will necessarily violate the tRP invariance. Local kernels, on the other hand,
generate divergences whose removal may or may not be possible. Hence, for the
time being, a compromise seems inevitable. We give up tRP invariance and
retain sDM invariance that allows regular nonlocal kernels. Just loosing tRP
invariance means loosing relativistic invariance. We are left with NR slow
motions in a distinguished frame: sDM is pointless. Also the space-time must
be nearly flat. That’s the Newtonian limit.

##  VI Newtonian hybrid classical-quantum gravity

When recapitulating the results of refs. Tilloy and Diósi (2016, 2017), we use
a particular approach. These works used the NR HSDE representation of hybrid
dynamics. Not for deduction but for comparison, we guide our derivation by the
HSDEs (42-44) that promised general relativistic postquantum gravity in sec.
V. We present the HSDEs of Newtonian hybrid theory first.

What is the closest NR dynamics to the HSDEs (42-V.1)? The matter Hamiltonian
with the hybrid coupling reads

$\hat{H}_{\mathrm{M}}[\Phi]=\hat{H}_{0}+\int\hat{\mu}\Phi dV$ (54)

where $\Phi$ is the Newton potential and $\hat{\mu}$ is the NR quantum field
of mass density. The quantum monitoring of $\hat{v}^{ij}$ corresponds to the
quantum monitoring of $\hat{\mu}(\mathbf{r})$ since the nonrelativistic limit
of $\hat{v}^{ij}$ is $\propto\hat{\mu}$. Hence the NR counterpart of the SDE
(42)

$\frac{d\hat{P}}{dt}\\!=\\!-\frac{i}{\hbar}[\hat{H}_{\mathrm{M}}[\Phi],\hat{P}]\\!+\\!\mathcal{D}_{\mathrm{Q}}\hat{P}\\!+\\!\frac{1}{\hbar}\mathbb{H}\\!\\!\int\\!\\!\left(\hat{\mu}\\!-\\!\langle\hat{\mu}\rangle\right)\\!\hat{P}wdV,$
(55)

with

$\mathcal{D}_{\mathrm{Q}}\hat{P}=-\frac{1}{8\hbar^{2}}\int\int\gamma_{\mathbf{r}\mathbf{s}}\left[\hat{\mu}(\mathbf{r}),[\hat{\mu}(\mathbf{s}),\hat{P}]\right]d\mathbf{r}d\mathbf{s}.$
(56)

The mesurement signal is of the standard form

$\langle\hat{\mu}\rangle+\widetilde{w},$ (57)

where
$\widetilde{w}(\mathbf{r},t)=\int\gamma^{\mathbf{r}\mathbf{s}}w(\mathbf{s},t)d\mathbf{s}$.
The covariant and contravariant components ($w,\widetilde{w}$) of the same
noise satisfy

$\displaystyle\mathbb{M}w(\mathbf{r},t)w(\mathbf{s},\tau)$ $\displaystyle=$
$\displaystyle\gamma_{\mathbf{r}\mathbf{s}}\delta(t-\tau)$
$\displaystyle\mathbb{M}\widetilde{w}(\mathbf{r},t)\widetilde{w}(\mathbf{s},\tau]$
$\displaystyle=$
$\displaystyle\hbar^{2}\gamma_{\mathbf{r}\mathbf{s}}\delta(t-\tau)$
$\displaystyle\mathbb{M}w(\mathbf{r},t)\widetilde{w}(\mathbf{s},\tau]$
$\displaystyle=$
$\displaystyle\hbar\delta(\mathbf{r}-\mathbf{s})\delta(t-\tau).$ (58)

Since gravity has no self-dynamics, $H_{\mathrm{G}}=0$, the backaction (43,44)
reduces to the Poisson equation sourced by the signal (57) and we can solve
it:

$\displaystyle\Phi(\mathbf{r},t)$ $\displaystyle=$ $\displaystyle\frac{4\pi
G}{\nabla^{2}}\left(\langle\hat{\mu}(\mathbf{r})\rangle_{t}+\widetilde{w}(\mathbf{r},t)\right)$
(59) $\displaystyle\equiv$
$\displaystyle\Phi_{\mathrm{mf}}(\mathbf{r},t)+\delta\Phi(\mathbf{r},t).$

The deterministic term $\Phi_{\mathrm{mf}}$ is the mean-field (semiclassical)
part, the stochastic term is a white-noise of correlation

$\mathbb{M}\delta\Phi(\mathbf{r},t)\delta\Phi(\mathbf{s},\tau)=\frac{4\pi
G}{\nabla^{2}_{\mathbf{r}}}\frac{4\pi
G}{\nabla^{2}_{\mathbf{s}}}\hbar^{2}\gamma_{\mathbf{r}\mathbf{s}}\delta(t-\tau).$
(60)

When $\Phi$ is fed back in eq. (55), the Hamiltonian
$\hat{H}_{\mathrm{M}}[\Phi]$ generates the Newtonian pair potential

$\hat{V}_{\mathrm{G}}=-\frac{G}{2}\int\int\frac{\hat{\mu}(\mathbf{r})\hat{\mu}(\mathbf{s})}{|\mathbf{r}-\mathbf{s}|}d\mathbf{r}d\mathbf{s}.$
(61)

Unlike the general relativistic $\hat{H}_{\mathrm{M}}[g]$, where $g$ is a
Wiener process, $\Phi$ is not, it is the time-derivative of a Wiener process.
The feedback of the white-noise term in $\hat{H}_{\mathrm{M}}[\Phi]$,
proportional to $\delta\Phi$, will contribute to a new decoherence term:

$\mathcal{D}_{\mathrm{Q}}^{\mathrm{fb}}\hat{P}=-\frac{1}{2\hbar^{2}}\int\\!\\!\\!\int\\!\\!\left(\frac{4\pi
G}{\nabla^{2}_{\mathbf{r}}}\frac{4\pi
G}{\nabla^{2}_{\mathbf{s}}}\gamma^{\mathbf{r}\mathbf{s}}\right)\\!\left[\hat{\mu}(\mathbf{r}),[\hat{\mu}(\mathbf{s}),\hat{P}]\right]\\!d\mathbf{r}d\mathbf{s}.$
(62)

This backaction makes a remarkable difference compared to the general
relativistic case in sec. V.2. The ambiguity of the D&D kernels can be removed
by the principle of least decoherence. Since
$\gamma^{\mathbf{r}\mathbf{s}}=\gamma_{\mathbf{r}\mathbf{s}}^{-1}$, the total
decoherence $\mathcal{D}_{\mathrm{Q}}+\mathcal{D}_{\mathrm{Q}}^{\mathrm{fb}}$
possesses a minimum when

$\displaystyle\gamma_{\mathbf{r}\mathbf{s}}$ $\displaystyle=$
$\displaystyle\frac{2\hbar G}{|\mathbf{r}-\mathbf{s}|},$
$\displaystyle\gamma^{\mathbf{r}\mathbf{s}}$ $\displaystyle=$
$\displaystyle-\frac{1}{8\pi\hbar G}\nabla^{2}\delta(\mathbf{r}-\mathbf{s}).$
(63)

Accordingly, the least decoherence reads

$\mathcal{D}_{\mathrm{Q}}^{\mathrm{DP}}=-\frac{G}{2\hbar}\int\int\frac{\left[\hat{\mu}(\mathbf{r}),[\hat{\mu}(\mathbf{s}),\hat{P}]\right]d\mathbf{r}d\mathbf{s}}{|\mathbf{r}-\mathbf{s}|}$
(64)

and the correlation of the gravitational fluctuations become

$\mathbb{M}\delta\Phi(\mathbf{r},t)\delta\Phi(\mathbf{s},\tau)=\frac{\hbar
G/2}{|\mathbf{r}-\mathbf{s}|}\delta(t-\tau).$ (65)

We obtain the HSDEs of the Newtonian NR postquantum gravity:

$\displaystyle\\!\\!\\!\\!\\!\frac{d\hat{P}}{dt}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!-\frac{i}{\hbar}[\hat{H}_{0}\\!+\\!\\!\hat{V}_{\mathrm{G}},\\!\hat{P}]\\!+\\!\\!\mathcal{D}_{\mathrm{Q}}^{\mathrm{DP}}\\!\hat{P}\\!+\\!\mathbb{H}\frac{1\\!\\!+\\!i}{\hbar}\\!\\!\\!\int\\!\\!\\!\left(\\!\hat{\mu}\\!-\\!\\!\langle\hat{\mu}\rangle\\!\right)\\!\hat{P}wdV$
(66) $\displaystyle\\!\\!\\!\\!\\!\Phi$ $\displaystyle=$
$\displaystyle\frac{4\pi
G}{\nabla^{2}}\langle\hat{\mu}\rangle-\frac{1}{2}w=\Phi_{\mathrm{mf}}-\frac{1}{2}w,$
(67)

where $\Phi_{\mathrm{mf}}$ is the mean-field (semiclassical) Newton potential,
and

$\mathbb{M}w(\mathbf{r},t)w(\mathbf{s},\tau)=\frac{2\hbar
G}{|\mathbf{r}-\mathbf{s}|}\delta(t-\tau).$ (68)

For point-like particles the theory is divergent, predicts kinetic energy
increase at infinite rate. Therefore $\hat{\mu}(\mathbf{r})$ must be
smoothened by a short length cutoff parameter, the only free parameter of the
theory (see Donadi  _et al._ (2021) for its experimental limit).

Observe that due to the simple structure of the Newtonian postquantum dynamics
the reduced dynamics of the quantized matter is autonomous. Take the
stochastic mean of both sides of eq. (66) then the following Lindblad master
equation is obtained for $\hat{\rho}_{\mathrm{Q}}=\mathbb{M}\hat{P}$:

$\frac{d\hat{\rho}_{\mathrm{Q}}}{dt}=-\frac{i}{\hbar}[\hat{H}_{0}+\hat{V}_{\mathrm{G}},\hat{\rho}_{\mathrm{Q}}]-\frac{G}{2\hbar}\int\\!\\!\\!\\!\int[\hat{\mu}(\mathbf{r}),[\hat{\mu}(\mathbf{s}),\hat{\rho}_{\mathrm{Q}}]]\frac{d\mathbf{r}d\mathbf{s}}{|\mathbf{r}-\mathbf{s}|}.$
(69)

The full HME, equivalent to the HSDE formalism (66-68), is derived in Appendix
B.

##  VII Remarks, Conclusions

The issues of hybrid dynamics relativistic extensions that secs. IV and V
claim are unsolved, were carefully discussed by the authors of refs. Oppenheim
_et al._ (2023a); Oppenheim (2023), highlighting some perspectives towards
solutions. These are assessed with certain reservation in ref. Tilloy (2018).
We add that the literature offers no support for hybrid constraints, little or
no support for renormalizability of relativistic effective field theories let
them be classical, quantum, or hybrid. Towards fixing infinities predicted by
relativistic Lindblad and Fokker–Planck equations, conclusive research is
missing even for the the simple special relativistic D&D in sec. IV.

Some additional details about the nonrelativistic ‘postquantum’ theory (sec.
VI) are to be recalled. It all started in foundations (reviewed in Bassi  _et
al._ (2013, 2023)), with a gravity-related nonrelativistic model of the
quantum-classical transition Diósi (1989) and a naiv formalism of relativistic
monitoring-plus-feedback Diósi (1990). Recognizing the difficulties of
relativistic monitoring, only the Newtonian limit of monitoring-plus-feedback
was briefly presented. Much later, the concept of postquantum gravity, called
a ‘conceptually healthier semiclassical theory’, was stated literally in
Tilloy and Diósi (2016): monitoring the quantized energy-momentum tensor
$\hat{T}_{ab}$ and its measured value fed back into the Einstein equation of
classical general relativity. After two and a half decades, this work and its
followup Tilloy and Diósi (2017) must still have adhered to the Newtonian
limit. The reason has remained the same: missing theory of relativistic
monitoring. The concrete technical obstacles are the D&D kernels that must be
time-local for Markovianity. If the suitably covariant kernels exist at all,
they generate divergences whose treatment is unknown. Without these
difficulties, the monitoring-plus-feedback form (equivalent to the hybrid
master equation form) of postquantum general relativity would have been a
straightforward step. Vice versa, if the hybrid master equation form of
postquantum gravity got rid of its difficulties with the D&D kernels, it would
contain a modul of relativistic quantum monitoring. This matches with the
assessment in ref. Tilloy (2018).

The pending issues of the recent proposal Oppenheim  _et al._ (2023a);
Oppenheim (2023) of postquantum gravity are the old difficulties that
knowingly hindered the relativistic extension of the Newtonian ‘forerunner’
Diósi (1990); Tilloy and Diósi (2016, 2017). The difficulties are rooted in
difficulties of Lindblad as well as of Fokker–Planck dynamics of relativistic
fields, which are due parts of postquantum gravity. Although these issues
might become fixed later, the contrary is equally likely: relativity and
Markovianity of decoherence (or diffusion) may turn out to be just
inconsistent Diósi (2022). The present author expects that the hybrid of
classical gravity and quantized matter is hiding more secrets already in the
Newtonian limit. We should continue to reveal them in the simple
nonrelativistic realm (available to laboratory tests, cf., e.g., ref. Folger
(2019)) before we would cross the bridge towards a certain postquantum general
relativity.

###### Acknowledgements.

I thank Isaac Layton, Jonathan Oppenheim, Andrea Russo and Antoine Tilloy for
illuminating discussions. This research was funded by the Foundational
Questions Institute and Fetzer Franklin Fund, a donor-advised fund of the
Silicon Valley Community Foundation (Grant No. FQXi-RFPCPW-2008), the National
Research, Development and Innovation Office (Hungary) “Frontline” Research
Excellence Program (Grant No. KKP133827), and the John Templeton Foundation
(Grant 62099).

##  Appendix A Deduction of HME (4)

We show that our canonical HME (4) with the D&D term (8) is the special case
of the general diffusive HME Oppenheim  _et al._ (2022, 2023b); Diósi (2023a,
b):

$\displaystyle\frac{d\hat{\rho}}{dt}$ $\displaystyle=$
$\displaystyle-i[\hat{H},\hat{\rho}]+2\mathbb{H}([\overline{G}_{\mathrm{CQ}}]^{n}_{\alpha}\hat{L}^{\alpha}\hat{\rho})_{,n}+\mathcal{D}\hat{\rho}$
(70) $\displaystyle\mathcal{D}$ $\displaystyle=$ $\displaystyle
D^{\mathrm{Q}}_{\beta\alpha}\Bigl{(}\hat{L}^{\alpha}\hat{\rho}\hat{L}^{\beta}-\mathbb{H}\hat{L}^{\beta}\hat{L}^{\alpha}\hat{\rho}\Bigr{)}+\tfrac{1}{2}\left(D_{\mathrm{C}}^{nm}\hat{\rho}\right)_{nm}$
(71)

where, compared to eq. (36) in Diósi (2023a), we assumed Hermitian Lindblad
generators $\hat{L}^{\alpha}$ and changed the upper/lower greek indices for
the lower/upper ones. This HME is valid for any classical subsystem, the
classical coordinates $x$ are not necessarily canonical. When the Lindblad
generators $\hat{L}^{\alpha}(x)$ are independent operators then minimum noise
is achieved if the positive D&D matrices $D^{\mathrm{Q}},D_{\mathrm{C}}$,
resp., are constrained by the matrix of backaction $G_{\mathrm{CQ}}$:

$G_{\mathrm{CQ}}\frac{1}{D^{\mathrm{Q}}}G_{\mathrm{CQ}}^{\dagger}=D_{\mathrm{C}}.$
(72)

Let us first identify the classical variables $x^{n}$ by our canonical ones.
Second, identify the Lindblad generators $\hat{L}^{\alpha}$ by our velocity
operators $\hat{v}^{n}$, the greek indices will become the latin ones
accordingly. Let us equate the backaction terms in (4) and (70):

$\mathbb{H}(\hat{v}^{n}\hat{\rho})_{,n}=-2\mathbb{H}([\overline{G}_{\mathrm{CQ}}]^{n}_{~{}m}\hat{v}^{n}\hat{\rho})_{,n}.$
(73)

They coincide if
$[\overline{G}_{\mathrm{CQ}}]^{n}_{m}=-\tfrac{1}{2}\delta^{n}_{m}$. The D&D
terms (8) and (71) coincide if $D^{\mathrm{Q}}_{nm}=\tfrac{1}{4}\gamma_{nm}$
and $D_{\mathrm{C}}^{nm}=\gamma^{nm}$. The said choices
$D_{\mathrm{C}},D^{\mathrm{Q}}$ and $G_{\mathrm{CQ}}$ satisfy the general
condition (72) of minimum noise.

##  Appendix B Derivation of HME from HSDEs (66-68)

It is incorrect to take the form $\hat{\rho}[\Phi]$ for the hybrid state since
$\Phi$ is a white noise. The correct form is $\hat{\rho}_{t}[\chi]$, i.e., the
configuration of classical gravity is represented by the Wiener process $\chi$
defined by $\Phi=d\chi/dt$. We define the hybrid density as follows:

$\hat{\rho}_{t}[\chi]=\mathbb{M}\hat{P}_{t}\delta[\chi-\chi_{t}]$ (74)

The differentials of both sides read

$d\hat{\rho}_{t}[\chi]=\mathbb{M}\left(d\hat{P}_{t}\delta[\chi-\chi_{t}]+\hat{P}_{t}d\delta[\chi-\chi_{t}]+d\hat{P}_{t}d\delta[\chi-\chi_{t}]\right)$
(75)

where the last term on the r.h.s. is the Ito correction to the Leibnitz rule.
According to Ito calculus, using the HSDEs (66,67) and the white-noise
correlation (68) yield

$\displaystyle d\hat{P}\\!$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}[\hat{H}_{0}+\hat{V}_{\mathrm{G}},\hat{P}]dt+\mathcal{D}_{\mathrm{Q}}^{\mathrm{DP}}\hat{P}dt$
(76)
$\displaystyle+\mathbb{H}\frac{1\\!+\\!i}{\hbar}\\!\\!\int\\!\\!\left(\hat{\mu}(\mathbf{r})\\!-\\!\langle\hat{\mu}(\mathbf{r})\rangle\right)\\!\hat{P}w(\mathbf{r},t)d\mathbf{r}dt$
$\displaystyle d\delta[\chi\\!-\\!\chi_{t}]\\!\\!$ $\displaystyle=$
$\displaystyle\\!-\\!\\!\int\\!\left(\Phi_{\mathrm{mf}}(\mathbf{r})\\!-\\!\tfrac{1}{2}w(\mathbf{r},t)\right)\\!\frac{\delta}{\delta\chi(\mathbf{r})}\delta[\chi\\!-\\!\chi_{t}]d\mathbf{r}dt$
(77)
$\displaystyle\\!+\\!\frac{1}{4}\\!\\!\int\\!\\!\\!\\!\\!\int\\!\\!\\!\frac{\hbar
G}{|\mathbf{r}\\!-\\!\mathbf{s}|}\frac{\delta^{2}}{\delta\chi(\mathbf{r})\delta\chi(\mathbf{s})}\delta[\chi\\!-\\!\chi_{t}]d\mathbf{r}d\mathbf{s}dt$
$\displaystyle d\hat{P}d\delta[\chi\\!-\\!\chi_{t}]\\!$ $\displaystyle=$
$\displaystyle\mathbb{H}(1\\!+\\!i)\int\\!\\!\\!\\!\\!\int\\!\\!\\!\frac{G}{|\mathbf{r}\\!-\\!\mathbf{s}|}\\!\left(\hat{\mu}(\mathbf{s})\\!-\\!\langle\hat{\mu}(\mathbf{s})\rangle\right)\hat{P}\times$
(78)
$\displaystyle\times\frac{\delta}{\delta\chi(\mathbf{r})}\delta[\chi\\!-\\!\chi_{t}]d\mathbf{r}d\mathbf{s}dt.$

Now we insert these three expressions into eq. (75), set $w=0$ since
$\mathbb{M}w=0$, and use the definition (74) of $\hat{\rho}[\chi]$, yielding,
after dividing both sides by $dt$:

$\displaystyle\frac{d\hat{\rho}[\chi]}{dt}=$
$\displaystyle-\frac{i}{\hbar}[\hat{H}_{0}+\hat{V}_{\mathrm{G}},\hat{\rho}[\chi]]+\mathcal{D}_{\mathrm{Q}}^{\mathrm{DP}}\hat{\rho}[\chi]$
(79)
$\displaystyle-\\!\\!\int\\!\Phi_{\mathrm{mf}}(\mathbf{r})\\!\frac{\delta\hat{\rho}[\chi]}{\delta\chi(\mathbf{r})}d\mathbf{r}\\!\\!+\frac{1}{4}\int\\!\\!\\!\\!\\!\int\\!\\!\\!\frac{\hbar
G}{|\mathbf{r}\\!-\\!\mathbf{s}|}\\!\frac{\delta^{2}\hat{\rho}[\chi]}{\delta\chi(\mathbf{r})\delta\chi(\mathbf{s})}d\mathbf{r}d\mathbf{s}$
$\displaystyle+\mathbb{H}(1\\!+\\!i)\\!\\!\int\\!\\!\\!\\!\\!\int\\!\\!\\!\frac{G}{|\mathbf{r}\\!-\\!\mathbf{s}|}\\!\left(\hat{\mu}(\mathbf{s})\\!-\\!\langle\hat{\mu}(\mathbf{s})\rangle\right)\frac{\delta\hat{\rho}[\chi]}{\delta\chi(\mathbf{r})}d\mathbf{r}d\mathbf{s}.$

The nonlinear terms on the r.h.s. cancel as they should and we write the HME
in the following form:

$\frac{d\hat{\rho}}{dt}=-\frac{i}{\hbar}[\hat{H}_{0}+\hat{V}_{\mathrm{G}},\hat{\rho}]+G\\!\int\\!\\!\\!\int\\!\left(-\frac{1}{2\hbar}[\hat{\mu}(\mathbf{r}),[\hat{\mu}(\mathbf{s}),\hat{\rho}]]+\mathbb{H}(1\\!-\\!i)\hat{\mu}(\mathbf{r})\frac{\delta\hat{\rho}}{\delta\chi(\mathbf{s})}+\frac{\hbar}{4}\frac{\delta^{2}\hat{\rho}}{\delta\chi(\mathbf{r})\delta\chi(\mathbf{s})}\right)\frac{d\mathbf{r}d\mathbf{s}}{|\mathbf{r}-\mathbf{s}|}.$
(80)

The HME yields ‘space-time’ diffusion (65) with
$\delta\Phi\\!=\\!d\chi/dt\\!-\\!\Phi_{\mathrm{mf}}$ and the mean-field
(semiclassical) gravity:

$\mathbb{M}\Phi(\mathbf{r})=\mathsf{tr}\\!\\!\int\\!\\!\frac{d\chi(\mathbf{r})}{dt}\hat{\rho}[\chi]d[\chi]=-G\\!\\!\int\\!\\!\frac{\langle\hat{\mu}(\mathbf{s})\rangle}{|\mathbf{r}-\mathbf{s}|}d\mathbf{s}=\Phi_{\mathrm{mf}}(\mathbf{r}).$
(81)

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