  The absence of radio pulsars with long periods has lead to the popular notion
of a high P ``death line.'' In the standard picture, beyond this boundary,
pulsars with low spin rates cannot accelerate particles above the stellar
surface to high enough energies to initiate pair cascades, and the pair
creation needed for radio emission is strongly suppressed. In this paper we
explore the possibility of another pulsar ``death line'' in the context of
polar cap models, corresponding to high magnetic fields B in the upper portion
of the period-period derivative diagram, a domain where few radio pulsars are
observed. The origin of this high B boundary, which may occur when B becomes
comparable to or exceeds $B_{\rm cr} = 4.4 \times 10^{13}$ Gauss, is also due
to the suppression of magnetic pair creation, but primarily because of
ineffective competition with magnetic photon splitting. Threshold pair creation
also plays a prominent role in the suppression of cascades. We present Monte
Carlo calculations of the pair yields in photon splitting/pair cascades which
show that, in the absence of scattering effects, pair production is effectively
suppressed, but only if all three modes of photon splitting allowed by QED are
operating in high fields. This paper describes the probable shape and position
of the new ``death line,'' above which pulsars are expected to be radio quiet,
but perhaps still X-ray and gamma-ray bright. The hypothesized existence of
radio-quiet sources finds dramatic support in the recent discovery of
ultra-strong fields in Soft Gamma-ray Repeaters and Anomalous X-ray Pulsars.
Guidelines for moderate to high B pulsar searches at radio wavelengths and also
in the soft and hard gamma-ray bands are presented.
