Varieties of instability of a boson field in an external potential

A relativistic scalar field is quantized in a one-dimensional ''box'' comprising two broad electrostatic potential wells. As the potential difference increases, the phenomena found long ago by Schiff, Snyder, and Weinberg in such a model occur: merging of mode frequencies and disappearance of the vacuum as a discrete state, followed by appearance of complex frequencies and unboundedness below of the total energy. However, a new effect appears for some values of the potential: The discrete vacuum (with the associated particle interpretation) reappears, but the energy remains unbounded below because some negative-norm modes have greater frequencies than some positive-norm modes. That is, a particle--antiparticle pair can have energy less than that of empty space. As the outer walls of the box approach infinity, this situation goes over into the boson Klein ''paradox,'' marked by nonuniqueness of the vacuum (spontaneous breaking of time-reversal symmetry) and coexistence of positive- and negative-norm continuum modes at the same frequency. These phenomena are of interest in connection with current work on pair creation in external gravitational fields, especially black holes