Abstract
Quantum Field Theories (QFTs) in Anti-de Sitter (AdS) spacetime are often strongly coupled when the radius of AdS is large, and few methods are available to study them. In this work, we develop a Hamiltonian truncation method to compute the energy spectrum of QFTs in two-dimensional AdS. The infinite volume of constant timeslices of AdS leads to divergences in the energy levels. We propose a simple prescription to obtain finite physical energies and test it with numerical diagonalization in several models: the free massive scalar field, ϕ4 theory, Lee-Yang and Ising field theory. Along the way, we discuss spontaneous symmetry breaking in AdS and derive a compact formula for perturbation theory in quantum mechanics at arbitrary order. Our results suggest that all conformal boundary conditions for a given theory are connected via bulk renormalization group flows in AdS.
Highlights
Coupled Quantum Field Theories (QFT) are challenging
Since the subgroup of the conformal group preserved by a flat boundary coincides with the isometries of Anti-de Sitter (AdS), any Conformal Field Theories (CFT) placed in AdS with isometry preserving boundary conditions will be related to a Boundary CFT (BCFT) by a Weyl transformation
The boundary condition generates a state in radial quantization, the so called Cardy state, which allows for a decomposition in terms of local operators of the bulk CFT [46]
Summary
Coupled Quantum Field Theories (QFT) are challenging. In this work, we are interested in UV complete QFTs defined as relevant deformations of free or solvable Conformal Field Theories (CFT). For example, if the QFT has a mass gap, we would like to determine the masses of the stable particles and their scattering amplitudes. A QFT on a compact spatial manifold like a sphere of radius R offers a new handle into the non-perturbative regime Energy spectrum as a function of R to interpolate from the perturbative small-R regime to the strongly coupled large-R limit.2 This is the basis of the Hamiltonian truncation approach to QFTd+1 placed on R × Sd, initiated in [5] for the case of 1+1 dimensions.. This parameter allows us to continuously connect the perturbative regime of small λwith the strongly coupled regime of large λ This is similar to the case mentioned above of QFT on R × Sd. The choice of the AdSd+1 background has two main advantages. We are especially proud of appendix B.1, where we derive a new simple formula for the order-λn correction to the eigenvalues of the Hamiltonian in perturbation theory
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