  Strongly coupled positronium, considered in its pseudoscalar sector, is
studied in the framework of relativistic quantum constraint dynamics. Case's
method of self-adjoint extension of singular potentials, which avoids explicit
introduction of regularization cut-offs, is adopted. It is found that, as the
coupling constant \alpha increases, the bound state spectrum undergoes an
abrupt change at the critical value \alpha = \alpha_c = 1/2. For \alpha >
\alpha_c, the mass spectrum displays, in addition to the existing states for
\alpha < \alpha_c, a new set of an infinite number of bound states concentrated
in a narrow band starting at mass W=0. In the limit \alpha going to \alpha_c
from above, these states shrink to a single massless state with a mass gap with
the rest of the spectrum. This state has the required properties to represent a
Goldstone boson and to signal a spontaneous breakdown of chiral symmetry. It is
suggested that the critical coupling constant \alpha_c be viewed as a possible
candidate for an ultra- violet stable fixed point of QED, with a distinction
between two phases, joined to each other by a first-order chiral phase
transition.
