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

Canonical quantization on scalar and Maxwell fields is studied for the stability of vacuum and construction of quantum dynamics in Anti-de Sitter spacetimes. The $ξRϕ^2$ term in the Lagrangian of scalar field with $ξ> 5/48$ (mass term just introduces shift of $ξ$) can cause the spectrum of the self-adjoint Hamiltonian with negative value not bounded from below. The degrees of freedom corresponding to the negative values are constrained system and are ambiguous in space direction and in commutation relation. The only other algebra accommodated in 3-dimensional space consistent with locality, the anti-commutation relation, is definite confronting the constrain, and trivialize the negative spectrum to be inactive and absent degrees of freedom. Thus only the physical bosons corresponding to the non-negative spectrum span the Hilbert space. For the Maxwell fields as another constrained system, the gauge invariant canonical energy momentum tensor straightforwardly leads to the gauge invariant non-negative Hamiltonian. The 2-dimensional independent dynamical degrees of freedom are quantized in temporal gauge. The energy momentum tensors for both quantized fields are renormalized to be finite at operator level, which guarantee the stable vacuum maximally symmetric and the particle Fock states the representation of the AdS symmetry SO(2,3). Adopting Weinberg's framework, the new covariant free quantum scalar field is reconstructed in the corresponding mode function space isomorphic to the free boson Hilbert space, allowing various local effective interaction terms among fields introduced. The Yang-Feldman equations relate the free fields and the interacting fields with the Green functions determined properly in the mode function space.