  We review the development of the most general analytic form of the cross
section, dependent upon the three polarization vectors of the beam, target and
recoil baryon, including all single, double and triple-polarization terms
involving 16 spin-dependent observables. We examine the different conventions
that have been used by different authors, and we present expressions that allow
the direct numerical calculation of any pseudoscalar meson photo-production
observables with arbitrary spin projections from the Chew-Goldberger-Low-Nambu
amplitudes. We use this numerical tool to clarify apparent sign differences
that exist in the literature. We also present analytic expressions that
determine the recoil baryon polarization, together with examples of their
potential use with quasi-4pi detectors to deduce observables. As an
illustration of the use of the consistent machinery presented in this review,
we carry out a multipole analysis of the gamma p -> K^+ Lambda reaction and
examine the impact of recently published polarization measurements. In fitting
multipoles, we use a combined Monte Carlo sampling of the amplitude space, with
gradient minimization, and find a shallow chi^2 valley pitted with a very large
number of local minima. This results in broad bands of multipole solutions that
are experimentally indistinguishable. While these bands have been noticeably
narrowed by the inclusion of new polarization measurements, many of the
multipoles remain very poorly determined, even in sign, despite the inclusion
of data on 8 different observables. We have compared multipoles from recent PWA
codes with our model-independent solution bands. The potential accuracy of
amplitudes that could be extracted from measurements of all 16 polarization
observables has been studied with mock data using the statistical variations
that are expected from ongoing experiments.
