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

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 \xi R\phi^2 term in the Lagrangian (mass can be introduced by shift of \xi), only when \xi \le 5/48, the quantized Hamiltonian is spontaneously non-negative and the vacuum is well defined. For \xi > 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 \xi\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.