  The Two-Body Dirac equations of constraint theory are of special interest not
only in view of applications for phenomenological calculations of mesonic
spectra but also because they avoid no-go theorems about relativistic
interactions. Furthermore, they provide a quantum mechanical description in a
manifestly Lorentz invariant way using the concept of a multi-time wave
function. In this paper, we place them into the context of the multi-time
formalism of Dirac, Tomonaga and Schwinger for the first time. A general
physical and mathematical framework is outlined and the mechanism which permits
relativistic interaction is identified. The main requirement derived from the
general framework is the existence of conserved tensor currents with a positive
component which can play the role of a probability density. We analyze this
question for a general class of Two-Body Dirac equations thoroughly and
comprehensively. While the free Dirac current is not conserved, it is possible
to find replacements. Improving on previous research, we achieve definite
conclusions whether restrictions of the function space or of the interaction
terms can guarantee the positive definiteness of the currents -- and whether
such restrictions are physically adequate. The consequences of the results are
drawn, with respect to both applied and foundational perspectives.
