Quantum approach to bound states in field theory

It is well known that (possibly nonunique) suitable field dynamics can be prescribed in spacetimes with timelike boundaries by means of appropriate boundary conditions. In [R. M. Wald, J. Math. Phys. 21, 2802 (1980)], Wald derived a conserved energy functional for each prescribed dynamics. This conserved energy is related to the positive self-adjoint extensions of the spatial part $A$ of the wave equation ${\partial }^2\Phi /\partial t^2=-A\Phi$ ($A$ may not be, in principle, essentially self-adjoint). This is quite surprising since the canonical energy is not conserved in these cases. In this paper, we rederive this energy functional from an action principle (with appropriate boundary terms) following [A. A. Saharian, Phys. Rev. D 69, 085005 (2004)] and consider field dynamics arising from nonpositive self-adjoint extensions of $A$. The spectrum of the resulting theory fails to be positive and unstable mode solutions for classical fields come to light. By studying fields in half-Minkowski spacetime, we illustrate that these unstable classical solutions come as a consequence of an inverted parabolic potential governing their dynamics. From the quantum mechanical point of view, this leads to an effective inverted harmonic oscillator at the boundary. We then explore these unstable modes behavior, as well as their instabilities, at the quantum level.