Classical Simulations of Quantum Field Theory in Curved Spacetime I: Fermionic Hawking-Hartle Vacua from a Staggered Lattice Scheme

TL;DR

This paper develops a numerical lattice approach to compute renormalized expectation values of fermionic quantum fields in curved spacetime, demonstrating applications like the Unruh effect and black hole vacua, paving the way for simulating interacting QFTs.

Contribution

It introduces a staggered lattice scheme combined with renormalization techniques to simulate free fermionic QFTs in curved spacetime, enabling future studies of interacting theories.

Findings

Successfully reproduces the Unruh effect in flat spacetime.

Calculates renormalized expectation values in black hole and de Sitter vacua.

Establishes a numerical framework for simulating QFTs in curved backgrounds.

Abstract

We numerically compute renormalized expectation values of quadratic operators in a quantum field theory (QFT) of free Dirac fermions in curved two-dimensional (Lorentzian) spacetime. First, we use a staggered-fermion discretization to generate a sequence of lattice theories yielding the desired QFT in the continuum limit. Numerically-computed lattice correlators are then used to approximate, through extrapolation, those in the continuum. Finally, we use so-called point-splitting regularization and Hadamard renormalization to remove divergences, and thus obtain finite, renormalized expectation values of quadratic operators in the continuum. As illustrative applications, we show how to recover the Unruh effect in flat spacetime and how to compute renormalized expectation values in the Hawking-Hartle vacuum of a Schwarzschild black hole and in the Bunch-Davies vacuum of an expanding universe described by de Sitter spacetime. Although here we address a non-interacting QFT using free fermion techniques, the framework described in this paper lays the groundwork for a series of subsequent studies involving simulation of interacting QFTs in curved spacetime by tensor network techniques.