Statistical Gravity Through Affine Quantization
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe paper proposes an approach to incorporate the effects of temperature into Einstein’s theory of general relativity by leveraging affine quantization and path integral Monte Carlo techniques. This method attempts to quantify thermal statistical effects on spacetime, offering insights into low-temperature quantum effects. The paper ties general relativity, statistical physics, and quantum mechanics into a cohesive framework. The proposal to use affine quantization for addressing thermal statistical effects in spacetime is both original and ambitious. This adds a fresh perspective to the ongoing challenge of unifying general relativity with quantum mechanics.
1. The authors should try to expand the introduction and include recent developments on gravity-quantum mechanics interplay such as:
- "Observation of a phase space horizon with surface gravity water waves." Communications Physics 7.1 (2024): 165.
- “Castelvecchi, Davide. "Artificial black hole creates its own version of Hawking radiation." Nature 536.7616 (2016).”
2. Providing preliminary results from path integral Monte Carlo simulations or outlining computational challenges would strengthen the study's practical applicability.
3. Can you suggest experimental setups or astrophysical observations that could test the proposed temperature effects in spacetime.
4. How does the assumption of a flat metric tensor reconcile with the statement that temperature effects should influence the curvature of spacetime, and could a non-flat metric be more appropriate for capturing such effects within the framework of affine quantization?
5. Equations (20) and (21) are crucial for connecting the FEBB theory with classical thermodynamics. These should be discussed in more detail, with explicit steps to bridge the theoretical derivation and the resulting physical interpretations.
6. The statement regarding the freedom to choose β<0 is intriguing but ambiguous. The author should clarify the physical meaning and implications of a negative inverse temperature parameter and how it aligns with the rest of the theory.
Author Response
----------------------------------------------------------------------------------------------------
Reply to the First Referee
----------------------------------------------------------------------------------------------------
1. I followed the Referee suggestion of expanding the Introduction mentioning three recent
developments on gravity-quantum mechanics-statistical physics interplay.
Therefore I added:
"
Some recent developments on gravity-quantum physics-statistical physics interplay are for example:
On 15 August 2016 Jeff Steinhauer, an experimental physicist at the Technion-Israel Institute of
Technology in Haifa, has created an artificial black hole that seems to emit `Hawking radiation' on
its own, from quantum fluctuations that emerge from its experimental set-up \cite{Castelvecchi2016}.
In a Bose-Einstein condensate (BEC) of rubidium atoms he created an event horizon by
accelerating the atoms until some were traveling at more than $1 {\rm mm}/{\rm s}$ -- a supersonic
speed for the condensate. On one side of his acoustical event horizon, where the atoms move at
supersonic speeds, phonons became trapped. And when Steinhauer took pictures of the BEC, he found
correlations between the densities of atoms that were an equal distance from the event horizon but
on opposite sides. This demonstrates that pairs of phonons were entangled -- a sign that they
originated spontaneously from the same quantum fluctuation and that the BEC was producing Hawking
radiation.
Although quantum objects, like the proton, are not conventionally considered within a unified
physics framework that includes gravity, it is significant to note that it has recently been
discovered that these particles are more dense than massively compact objects like neutron stars.
Scientists at Jefferson Lab have measured pressures of 100 billion trillion trillion pascals at the
proton's core -- about ten times greater than the pressure inside neutron stars. These are the same
pressure gradients computed by Haramein et al. \cite{Haramein2023}, from first-principle
considerations alone, on quantum vacuum fluctuations and gravitational effects at the proton scale,
suggesting spacetime curvature alone is the source of mass and force.
The measurements by the Jefferson Lab team were achieved through innovative experiments that use
pairs of photons to simulate gravitational interactions, allowing researchers to map the internal
forces and pressures of the proton for the first time. The team found two distinct regions of
specific pressures, like two phases within the proton, which was nearly exactly described in the
study ``The Origin of Mass and the Nature of Gravity'' \cite{Haramein2023} by Haramein and the
research
team at the International Space Federation. A perfect example of how empirical research can reveal
when a theory is on the right track, like when the Muonic measurements of the proton radius
confirmed Haramein's prediction of a smaller proton radius than what was expected by the Standard
Model, and now experimental data is revealing that, indeed, the mass-energy and pressures within the
proton are sufficient that spacetime curvature must be a major consideration in the mass and binding
forces that are observed for nucleons.
In 1974, Stephen Hawking predicted that quantum effects in the proximity of a black hole lead to the
emission of particles and black hole evaporation. At the very heart of this process lies a
logarithmic phase singularity which leads to the Bose-Einstein statistics of Hawking radiation. An
identical singularity appears in the elementary quantum system of the inverted harmonic oscillator.
In Ref. \cite{Rozenman2024} the authors report the observation of the onset of this logarithmic
phase singularity emerging at a horizon in phase space and giving rise to a Fermi-Dirac
distribution. For this purpose, they utilize surface gravity water waves and freely propagate an
appropriately tailored energy wave function of the inverted harmonic oscillator to reveal the phase
space horizon and the intrinsic singularities. Due to the presence of an amplitude singularity in
this system, the analogous quantities display a Fermi-Dirac rather than a Bose-Einstein
distribution.
In this work we try to carry to the extreme consequences the belief that the Wick rotation, that
allows to go back and forth between quantum physics and statistical physics in the non-relativistic
theory, will continue to do so also for the (quantum) general relativity theory. This belief is
based only on the grounds of a mathematical extension but allows us to reach the amusing physical
interpretation of a fabric of space-time (the main actor of the new theory) which has thermal
fluctuations itself. We will in fact introduce an effective temperature (field) and we will show its
relationship with the absolute temperature of thermodynamics in the non-quantum non-relativistic
limits.
"
2. I added the following on page 4:
"
The real metric fields $\{g\}$ take the values $\{g\}(x)$ on each site of a periodic
$4$-dimensional lattice of lattice spacing $a$, the ultraviolet cutoff, spatial periodicity
$L=Na$ and temporal periodicity $\beta=N_0a$. The metric path is a closed loop on a $4$-dimensional
surface of a $5$-dimensional $\beta$-cylinder.
We are interested in reaching the continuum limit by taking $L=Na$ fixed and letting
$N\to\infty$ at fixed volume $L^3$. As for the temporal periodicity we can
initially treat it on the same footing of the spatial periodicity taking $\beta=L$ and $N_0=N$.
On the other hand we can distinguish the two periodicities allowing the absolute
temperature $\tilde{T}=1/\tilde{k}_B\beta$ to vary so that the number of discretization
points for the imaginary time interval $[0,\beta[$ will be $N_0=\beta/a$.
The PIMC simulation (or computer experiment) may use the Metropolis algorithm
\cite{Kalos-Whitlock,Metropolis} to calculate the discretized version of Eq. (\ref{eq:EV})
which is a $10N^3N_0$ multidimensional integral. So clearly the computational resources
needed to perform a given simulation depends critically on $a$. We may use natural Planck units
$c = \hbar = k_B = 1$. The simulation is started from any initial condition.
One MC step consists in a random displacement of each one of the $10N^3N_0$
independent components of $\{g\}$ as follows
\bq
g^{\mu\nu}\rightarrow g^{\mu\nu}+(2\eta-1)\delta,
\eq
where $\eta$ is a uniform pseudo random number in $[0,1]$ and $\delta$ is the amplitude of the
displacement. Each one of these $10N^3N_0$ moves is accepted if
$\exp(-\upsilon\Delta S)>\eta$ where $\Delta S$ is the change in the
action due to the move and rejected otherwise. The amplitude $\delta$ is chosen in such a way
to have acceptance ratios as close as possible to $1/2$ and is kept constant during the
evolution of the simulation. One simulation consists of a
large number of $p$ steps. The statistical error on the average $\langle{\cal O}\rangle$ will then
depend on the correlation time, $\tau_{\cal O}$, necessary to decorrelate
the property ${\cal O}$ and will be determined as
$\sqrt{\sigma_{\cal O}^2\tau_{\cal O}/[p10N^3N_0]}$, where
$\sigma_{\cal O}^2$ is the intrinsic variance for ${\cal O}$.
Our estimate of the path integrals will be generally subject
to three sources of numerical uncertainties: The one due to the {\sl statistical error},
the one due to the {\sl spacetime discretization}, and the one due to the
{\sl finite-size effects}. Of these, the statistical
error scales like $P^{-1/2}$ where $P=p10N^3N_0$ is the computer time, the discretization of
spacetime is responsible for the distance from the continuum limit (which corresponds to a
lattice spacing $a\to 0$), and the finite-size effects stems from the necessity to approximate
the infinite spacetime system with one in a hypertorus of volume $L^3\beta$.
"
3. these were already suggested in the Introduction
4. Affine quantization is relevant for making more precise the definition of the
constant \upsilon. But this was already stated in the previous version
5. I added the following phrase after Eq. (21):
"
Our Eq. (\ref{eq:nq-nr}) makes sense since
infinity minus infinity is an indeterminate form and can be equal to a finite result.
"
6. I rephrased that sentence as follows:
"
But thanks to the circular boundary conditions around the imaginary time we are free to choose
$\beta<0$. And this should be the case (see footnote \ref{fn}) since from the trace of Einstein
field equations follows, in imaginary time, $-R=\kappa T^\mu_\mu>0$, i.e. $R<0$.
"
I hope that the new version will be suitable for publication in Quantum Reports.
Reviewer 2 Report
Comments and Suggestions for AuthorsPlease see the attached pdf.
Comments for author File: 
Comments.pdf
Author Response
----------------------------------------------------------------------------------------------------
Reply to the Second Referee
----------------------------------------------------------------------------------------------------
1. I do not have yet performed any computation on this new theory. In this first manuscript I had in
mind to describe the theory only. But I added the following on page 4:
"
The real metric fields $\{g\}$ take the values $\{g\}(x)$ on each site of a periodic
$4$-dimensional lattice of lattice spacing $a$, the ultraviolet cutoff, spatial periodicity
$L=Na$ and temporal periodicity $\beta=N_0a$. The metric path is a closed loop on a $4$-dimensional
surface of a $5$-dimensional $\beta$-cylinder.
We are interested in reaching the continuum limit by taking $L=Na$ fixed and letting
$N\to\infty$ at fixed volume $L^3$. As for the temporal periodicity we can
initially treat it on the same footing of the spatial periodicity taking $\beta=L$ and $N_0=N$.
On the other hand we can distinguish the two periodicities allowing the absolute
temperature $\tilde{T}=1/\tilde{k}_B\beta$ to vary so that the number of discretization
points for the imaginary time interval $[0,\beta[$ will be $N_0=\beta/a$.
The PIMC simulation (or computer experiment) may use the Metropolis algorithm
\cite{Kalos-Whitlock,Metropolis} to calculate the discretized version of Eq. (\ref{eq:EV})
which is a $10N^3N_0$ multidimensional integral. So clearly the computational resources
needed to perform a given simulation depends critically on $a$. We may use natural Planck units
$c = \hbar = k_B = 1$. The simulation is started from any initial condition.
One MC step consists in a random displacement of each one of the $10N^3N_0$
independent components of $\{g\}$ as follows
\bq
g^{\mu\nu}\rightarrow g^{\mu\nu}+(2\eta-1)\delta,
\eq
where $\eta$ is a uniform pseudo random number in $[0,1]$ and $\delta$ is the amplitude of the
displacement. Each one of these $10N^3N_0$ moves is accepted if
$\exp(-\upsilon\Delta S)>\eta$ where $\Delta S$ is the change in the
action due to the move and rejected otherwise. The amplitude $\delta$ is chosen in such a way
to have acceptance ratios as close as possible to $1/2$ and is kept constant during the
evolution of the simulation. One simulation consists of a
large number of $p$ steps. The statistical error on the average $\langle{\cal O}\rangle$ will then
depend on the correlation time, $\tau_{\cal O}$, necessary to decorrelate
the property ${\cal O}$ and will be determined as
$\sqrt{\sigma_{\cal O}^2\tau_{\cal O}/[p10N^3N_0]}$, where
$\sigma_{\cal O}^2$ is the intrinsic variance for ${\cal O}$.
Our estimate of the path integrals will be generally subject
to three sources of numerical uncertainties: The one due to the {\sl statistical error},
the one due to the {\sl spacetime discretization}, and the one due to the
{\sl finite-size effects}. Of these, the statistical
error scales like $P^{-1/2}$ where $P=p10N^3N_0$ is the computer time, the discretization of
spacetime is responsible for the distance from the continuum limit (which corresponds to a
lattice spacing $a\to 0$), and the finite-size effects stems from the necessity to approximate
the infinite spacetime system with one in a hypertorus of volume $L^3\beta$.
"
2. No, in fact already in the previous version I felt the need to let the temperature
become a field depending on space only. See the paragraph:
"
In Eq. (\ref{eq:EV}) we assumed a constant temperature throughout the whole accessible spacetime.
This can be a too restrictive condition and it could be necessary to think about a temperature
scalar field $\tilde{T}(x)$ which gives the value of the temperature in a neighborhood of a given
event $x$. Actually we are bound to choose $\tilde{T}(\xx)$ as the temperature in a neighborhood of
a given spatial point $\xx$ since we must require $x^0\in [0,\beta(\xx)[$. And the temperature field
$\tilde{T}(\xx)=1/\tilde{k}_B\beta(\xx)$ could be available experimentally.
"
3. I followed the Referee suggestion of expanding the Introduction mentioning three recent
developments on gravity-quantum mechanics-statistical physics interplay.
Therefore I added:
"
Some recent developments on gravity-quantum physics-statistical physics interplay are for example:
On 15 August 2016 Jeff Steinhauer, an experimental physicist at the Technion-Israel Institute of
Technology in Haifa, has created an artificial black hole that seems to emit `Hawking radiation' on
its own, from quantum fluctuations that emerge from its experimental set-up \cite{Castelvecchi2016}.
In a Bose-Einstein condensate (BEC) of rubidium atoms he created an event horizon by
accelerating the atoms until some were traveling at more than $1 {\rm mm}/{\rm s}$ -- a supersonic
speed for the condensate. On one side of his acoustical event horizon, where the atoms move at
supersonic speeds, phonons became trapped. And when Steinhauer took pictures of the BEC, he found
correlations between the densities of atoms that were an equal distance from the event horizon but
on opposite sides. This demonstrates that pairs of phonons were entangled -- a sign that they
originated spontaneously from the same quantum fluctuation and that the BEC was producing Hawking
radiation.
Although quantum objects, like the proton, are not conventionally considered within a unified
physics framework that includes gravity, it is significant to note that it has recently been
discovered that these particles are more dense than massively compact objects like neutron stars.
Scientists at Jefferson Lab have measured pressures of 100 billion trillion trillion pascals at the
proton's core -- about ten times greater than the pressure inside neutron stars. These are the same
pressure gradients computed by Haramein et al. \cite{Haramein2023}, from first-principle
considerations alone, on quantum vacuum fluctuations and gravitational effects at the proton scale,
suggesting spacetime curvature alone is the source of mass and force.
The measurements by the Jefferson Lab team were achieved through innovative experiments that use
pairs of photons to simulate gravitational interactions, allowing researchers to map the internal
forces and pressures of the proton for the first time. The team found two distinct regions of
specific pressures, like two phases within the proton, which was nearly exactly described in the
study ``The Origin of Mass and the Nature of Gravity'' \cite{Haramein2023} by Haramein and the
research
team at the International Space Federation. A perfect example of how empirical research can reveal
when a theory is on the right track, like when the Muonic measurements of the proton radius
confirmed Haramein's prediction of a smaller proton radius than what was expected by the Standard
Model, and now experimental data is revealing that, indeed, the mass-energy and pressures within the
proton are sufficient that spacetime curvature must be a major consideration in the mass and binding
forces that are observed for nucleons.
In 1974, Stephen Hawking predicted that quantum effects in the proximity of a black hole lead to the
emission of particles and black hole evaporation. At the very heart of this process lies a
logarithmic phase singularity which leads to the Bose-Einstein statistics of Hawking radiation. An
identical singularity appears in the elementary quantum system of the inverted harmonic oscillator.
In Ref. \cite{Rozenman2024} the authors report the observation of the onset of this logarithmic
phase singularity emerging at a horizon in phase space and giving rise to a Fermi-Dirac
distribution. For this purpose, they utilize surface gravity water waves and freely propagate an
appropriately tailored energy wave function of the inverted harmonic oscillator to reveal the phase
space horizon and the intrinsic singularities. Due to the presence of an amplitude singularity in
this system, the analogous quantities display a Fermi-Dirac rather than a Bose-Einstein
distribution.
In this work we try to carry to the extreme consequences the belief that the Wick rotation, that
allows to go back and forth between quantum physics and statistical physics in the non-relativistic
theory, will continue to do so also for the (quantum) general relativity theory. This belief is
based only on the grounds of a mathematical extension but allows us to reach the amusing physical
interpretation of a fabric of space-time (the main actor of the new theory) which has thermal
fluctuations itself. We will in fact introduce an effective temperature (field) and we will show its
relationship with the absolute temperature of thermodynamics in the non-quantum non-relativistic
limits.
"
I hope that the new version will be suitable for publication in Quantum Reports.
Round 2
Reviewer 2 Report
Comments and Suggestions for AuthorsThe author explained well, and responded to my queries/comments satisfactorily. Therefore, I recommend its publication.
