Quantum Effective Dynamics and Stability of Vacuum in Anti-de Sitter Spacetimes

TL;DR

The paper addresses vacuum stability of effective quantum field theories in anti-de Sitter spacetime by performing canonical quantization for a non-minimally coupled real scalar field and the Maxwell field. It employs ghost states to neutralize negative-energy modes when $\xi$ places the system outside positive-extension regimes, and, where possible, positive self-adjoint extensions to ensure a non-negative Hamiltonian, followed by a BPHZ-like renormalization of the energy–momentum tensor $T^{\mu\nu}_R$. For the Maxwell field, gauge invariance yields a non-negative, ghost-free Hamiltonian, with renormalization producing finite, maximally symmetric vacua. The results establish a consistent, globally defined Hilbert space and observables in AdS, laying groundwork for backreaction analyses and holographic implications through stable QEFT dynamics in curved spacetime.

Abstract

We investigate the details of the canonical quantization of effective quantum field theories in anti-de Sitter spacetime, emphasizing the stability of the quantum vacuum. We take the scalar and Maxwell fields as examples. For the non-minimally coupled massless real scalar field with ξRφ^2 term in the Lagrangian (mass can be introduced by shift of ξ), only when ξ\le 5/48, the quantized Hamiltonian is spontaneously non-negative and the vacuum is well defined. For ξ> 5/48, one has to assign the negative energy spectrum as that of the ghost particles, introducing anti-commutation relations to make the corresponding part of the Hamiltonian trivial, ensuring the Hamiltonian non-negative and the vacuum (and the Hilbert space) well defined. This method of ghost states is applicable once the proper radial boundary conditions guarantee the Hamiltonian self-adjoint. The resulting dynamics can be compared with those resulting from the positive self-adjoint extensions when the latter is available for ξ\le 9/48. For the Maxwell fields, the gauge invariant canonical energy momentum tensor straightforwardly leads to the gauge invariant non-negative Hamiltonian (well-defined vacuum). Hence the redundant gauge degree of freedom is irrelevant, and the 2-dimensional dynamical degrees of freedom are quantized in a concrete, e.g., temporal gauge. The energy momentum tensors for both quantized fields are renormalized to be finite at operator level, which renders the stable vacuum maximally symmetric. The back-reactions to the background spacetime by excited states via the semi-classical Einstein equations are also discussed.