Building spacetime from effective interactions between quantum fluctuations

TL;DR

This work investigates how spacetime and general relativity can emerge from effective interactions among quantum fluctuations in vacuum. It introduces an effective quantum model where each fluctuation is described by a Gaussian probability distribution in spacetime, with a momentum $P_o^{\mu}$ that drives a biased random walk; the ensemble average defines a spacetime metric $g_{\mu\nu}$ and reproduces geodesic motion through an emergent geodesic equation. A central result is that, under vacuum boundary conditions and the stated assumptions, the emergent metric is Ricci-flat ($R_{\mu\nu}=0$), linking microscopic quantum fluctuations to macroscopic GR in the large-scale limit. The framework also shows how boundary conditions can introduce nontrivial spacetime features such as frame-dragging, while remaining compatible with asymptotically flat configurations. This approach provides a concrete bridge between detailed quantum interactions and the effective large-scale theory, suggesting directions for extending to nonzero curvature and exploring connections with gauge/gravity duality and related many-body constructions.

Abstract

We describe how a model of effective interactions between quantum fluctuations under certain assumptions can be constructed in a way so that the large-scale limit gives an effective theory that matches general relativity in vacuum regions. This is an investigation of a possible scenario of spacetime emergence from quantum interactions directly in the spacetime, and of how effective quantum behaviour might provide a useful link between detailed properties of quantum interactions and general relativity. The quantum fluctuations are assumed to entangle sufficiently for a cohesive spacetime to form, so that their effective properties can be described relative to a D-dimensional reference frame. To obtain the desired features of a smooth metric with a vanishing Ricci tensor, the quantum fluctuations are modelled as Gaussian probability distributions, with a shape set relative to the interactions coming from the surroundings. At small scales, the propagation through the spacetime is modelled by a Gaussian random walk.

Figures (1)

• Figure 1: In each graph we see a surface $f(x)$ with a Gaussian distribution on it (coloration). To the left, the surface has $\partial_x^2 f(x)=0$ and follows the equation $f(x)=x$. The distribution is centred around $x=0.5$ and has standard deviation $\sigma=1/3$. To the right, the surface follows the equation $f(x)=x^2+0.5$, and the distribution has $\sigma=1/2$. Below each surface is an illustration of the distribution as seen from the $x$-axis (for clarity, these are not normalised). To the left, the Gaussian distribution on the surface is a Gaussian distribution also relative to the $x$-axis, but to the right the distribution is distorted. It is clearly visible that the Gaussian distribution on the bent curve translates into a non-Gaussian function relative to the $x$-axis. In the graph to the right, the mean value of $x$ is no longer $0.5$. Instead, $\langle x\rangle<0.5$. In addition, with an increase in $\sigma$ over time ($\partial_t\sigma>0$), the graph to the right is characterised by $\partial_t \langle x\rangle<0$, so that $\langle x\rangle$ moves to lower values. In the same scenario, the graph to the left has $\langle x\rangle=0.5$, $\partial_t \langle x\rangle=0$. What we want to illustrate with this picture is that a particle that moves on the surface according to a Gaussian random walk will have a position $\langle x(t)\rangle$ that depends on the shape of the surface. In this $1d$ example, $\partial_x^2 f\neq0$ causes a shift in $\langle x(t)\rangle$. In general, it is $\partial_\rho g_{\mu\nu}\neq0$ that causes this type of effect. For example, the moving particle can be a quantum fluctuation that starts out at $x=0.5$ with a Gaussian probability distribution for its position that is characterised by a $\sigma_x$ at a scale small enough for $\partial_x f$ to be well approximated by a constant. Relative to $\{x^\mu\}$, the particle will be subject to an interaction rate $f_t=\partial_x f$ from its environment. After $n$ steps in a Gaussian random walk with variance $\sigma_x^2$, the variance of the position distribution will be $\sigma^2=n\sigma_x^2$, and in that sense the $\langle x\rangle$ will evolve in time. Note that the evolution of $\langle x\rangle$ only takes place relative to $\{x^\mu\}$. The centre point of the distribution as seen from the surface $f(x)$ remains constant (at $x=0.5$ in the graphs).