Canonical Form Factors and Current Commutation Relations

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Abstract
Starting with the vector (or axial vector) currents jμ and the momentum operators Pν, we define the canonical operator j̃, such that instead of the four covariantly transforming components of j, we have a scalar j̃0 and three other components j̃ undergoing Wigner rotations under Lorentz transformations. We first give a construction of j̃ explicitly in terms of j and P. But, since the transformation properties are not quite the most convenient ones, a subsequent generalized definition, leading to a convenient canonical parametrization of the matrix elements of j̃, is introduced. We then study the physical significances of the canonical form factors thus obtained. For vector currents j̃v the transformation properties correspond to a separation of the physical charge (j̃0) and magnetic (j̃) form factors in any frame (and not only in Breit frame as for j). For nonconserved axial currents we relate the matrix elements of j̃0A with mass‐difference effects and express the partial conservation condition in terms of the canonical form factors. We then study in detail the application of our formalism to the limiting case of infinite momentum and small momentum‐exchange, as often introduced in the study of current algebras. Next we give explicitly the canonical form factors for photoproduction processes. In the last section we study the possibility of constructing a canonical spin operator directly in terms of the vector and axial vector charges and the consequences for the ``inner orbital'' contribution to be added to obtain the total spin of a composite particle. Some useful formulas are collected in Appendixes A and B.