The interrelation between the 4D and 3D forms of the Bethe–Salpeter equation (BSE) with a kernel [Formula: see text] which depends on the relative four-momenta, [Formula: see text] orthogonal to Pμ is exploited to obtain a hadron–quark vertex function of the Lorentz-invariant form [Formula: see text]. The denominator function [Formula: see text] is universal and controls the 3D BSE, which provides the mass spectra with the eigenfunctions [Formula: see text]. The vertex function, directly related to the 4D wave function Ψ which satisfies a corresponding BSE, defines a natural off-shell extension over the whole of four-momentum space, and provides the basis for the evaluation of transition amplitudes via appropriate quark-loop digrams. The key role of the quantity [Formula: see text] in this formalism is clarified in relation to earlier approaches, in which the applications of this quantity had mostly been limited to the mass shell (q · P = 0). Two applications (fP values for [Formula: see text] and Fπ for π0 → γγ) are sketched as illustrations of this formalism, and attention is drawn to the problem of complex amplitudes for bigger quark loops with more hadrons, together with the role of the [Formula: see text] function in overcoming this problem.

The 'Salpeter Equation ' which has long been known as the 3D version of the 4D Bethe-Salpeter Equation under the Instantaneous Approximation, has a well-defined rationale that stems from the half-century old Markov-Yukawa Transversality Principle (M Y T P ) which not only effects an exact 3D reduction from the original (4D) BS form, but also provides an equally exact reconstruction of the 4D BS amplitude in terms of 3D ingredients. The second aspect which is new, opens up a vista of applications to transition amplitudes as 4D loop integrals.

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