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.
Highlights
While it is generally expected that black holes will radiate and evaporate [12, 27, 52] in the presence of a quantum field, the details remain murky
Our target in this study will be the Hawking-Hartle vacuum, which describes a (Killing) horizon in thermal equilibrium with its environment. In spacetimes containing such Killing horizons, the Hawking-Hartle vacuum is, in a free theory, the unique state that is both stationary and possessed of a Hadamard function with no divergences in x independently of x. That such a state appears thermal to certain observers is the essential reason for the Unruh effect, the inflationary power spectrum, and black hole radiance, arguably the “poster children” of quantum field theory in curved spacetime
We could make a Jordan-Wigner transformation [30] to exactly map the lattice Hamiltonian (55) and lattice Hadamard functions (57) to expressions in terms of Pauli matrices while retaining their local character. This enables simulations of the ground state of (55) and other related states of interest using matrix product state techniques, as we explore in Ref. [35] in the context of simulating interacting quantum field theory (QFT)
Summary
While it is generally expected that black holes will radiate and evaporate [12, 27, 52] in the presence of a quantum field, the details remain murky. In a quantum field theory, the expectation value Tμν of the stress-energy tensor Tμν in the Hawking-Hartle vacuum is formally infinite even for free fields, and to extract a meaningful finite number it needs to be renormalized This same problem occurs for any operator with quadratic terms in the fields. The second develops the continuum problem to be numerically solved: it defines the free Dirac field in a curved two-dimensional spacetime, explicates its canonical quantization, and details how point-splitting regularization and Hadamard renormalization allow quadratic expectation values to be defined. It introduces examples of spacetimes with a bifurcate Killing horizon and defines the HawkingHartle vacuum. The fourth section uses the above method to find Hadamard-renormalized expectation values in the four example cases (i)-(iv) previously listed
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