Computational quantum field theory for fermion pair creation in 2-dimensional curved spacetimes

TL;DR

The paper tackles curvature-induced fermion pair creation in a curved 1+1D spacetime by extending Computational Quantum Field Theory (CQFT) to curved backgrounds. It recasts the Dirac equation on a curved manifold using vielbeins and the spin connection, producing a Hermitian Hamiltonian via $\chi=\alpha^{-1/4}\psi$ and a unitary split-operator evolution on a lattice. Applying this to a Gaussian curvature bump with ds^2 = $\alpha(\xi)$ d$\tau$^2$ - d$\xi$^2/ $\alpha(\xi)$ and $\alpha(\xi)=1-\beta e^{-\xi^2/r_0^2}$, the study shows curvature alone can drive quench-induced vacuum excitation and fermion–antifermion production relative to Minkowski modes, with real-time, spatially resolved observables such as densities and spectra. The framework sets the stage for including electromagnetic backgrounds and gravitational backreaction, enabling more realistic investigations of particle creation in curved spacetimes and related analogue-gravity systems.

Abstract

Similarly to the well-known phenomenon of particle / anti-particle pair production in strong electromagnetic fields (the Schwinger effect), the naïve matter field vacuum state can be excited by time-dependent, curved spacetime geometries. This gravitational pair creation corresponds to tunnelling out of a false vacuum. In this work, we study this non-perturbative process using a spacetime resolved numerical approach in the interaction picture. To achieve this, we extend the framework of Computational Quantum Field Theory (CQFT), which allows for efficient numerical time evolution of quantum fields, to spin-$1/2$ fermions in curved spacetime. Using this extended framework, we investigate vacuum excitation of a Dirac field induced by a spacetime-curvature quench. In particular, we evolve the fermionic Minkowski vacuum in a $1\!+\!1$-dimensional idealized curved spacetime characterized by a localized ``curvature bump'' generated by a smooth, localized Gaussian deformation of flat spacetime. Vacuum excitation is quantified by computing the fermion--antifermion pair numbers defined with respect to the basis corresponding to flat-spacetime (Minkowski) which is the asymptotic metric corresponding to an observor at infinity. We analyze how the excitation depends on the strength and spatial extent of the curvature deformation and discuss the numerical implementation of CQFT in curved backgrounds. While the post-quench geometry considered here is static and no electromagnetic field is included, the present work establishes a foundation for future investigations of particle creation in genuinely time-dependent curved spacetimes and in the presence of electromagnetic backgrounds.

Figures (7)

• Figure 1: Number of created fermion--antifermion pairs for a Sauter step given by Eq. \ref{['sauter_step']}, with $E_0=1/4$, and $\omega=0.1$. The upper panel shows the time evolution over a duration of $200 \lambda$, while the lower panel is a zoom of the late-time region. The obtained asymptotic number is $N \simeq 7.4 \times 10^{-6}$. The calculation is performed using CQFT on a lattice of width $100\,\lambda$ with $2^{11}$ sites and time step $\delta t = 0.001 \lambda$ increasing the number of lattice sites or reducing the time step does not change the curves at the resolution shown. The asymptotic number of created pairs agrees with the asymptotic adiabatic particle number ildertonadia.
• Figure 2: Number density of of created fermion--antifermion pairs (upper panel) and their momentum spectrum (lower panel) at three times: $\tau=0.1\,\lambda$ (dotted line), $\tau=1.3\,\lambda$ (dashed line), and $\tau=10\,\lambda$ (solid black line), where $\lambda$ is the electron Compton wavelength. The Ricci scalar $R(\xi)$, Eq. \ref{['ricci']} corresponding to the spacetime curvature given by Eqs. \ref{['metricschwa']} and \ref{['defo']} with $\beta=\tfrac{1}{2}$ is shown in the upper panel by the solid gray line. Calculations use $N=2^{11}$ lattice sites and a time step $\delta\tau=0.01\,\lambda$.
• Figure 3: Number of created pairs in the case of a Gaussian deformation of width $r=1 \lambda$ and lattice width $\Lambda = 100 \lambda$ for different lattice spacings corresponding to $N=2^9$ sites (dotted line), $N=2^{10}$ (dashed line), $N=2^{11}$ (solid line). For greater values of $N$, the curves become indistinguishable in the plotting resolution.
• Figure 4: Number growth of created fermion-antifermion pairs for different widths of the Gaussian curvature Eq. \ref{['defo']}. For wider curvatures, the production reaches saturation at later times and for greater numbers of created pairs. The dotted line corresponds to $r_0=1 \lambda$, the dash-dotted line corresponds to $r_0=\sqrt{2} \lambda$ and the solid line corresponds to $r_0=\sqrt{3} \lambda$. The numbers are calculated on a lattice of width $100 \lambda$ and of $N = 2^{11}$ sites.
• Figure 5: Number growth of created fermion-antifermion pairs for different strengths of the Gaussian curvature with $r_0=1 \lambda$. For stronger curvatures, more pairs are created and saturation is reached for greater numbers. The dotted line corresponds to $\beta=0.5$, the dash-dotted line corresponds to $\beta = 0.7$, the solid line corresponds to $\beta = 0.8$ and the dotted line corresponds to $\beta = 0.9$.