Uniqueness of the Interaction Involving Spin-32Particles

The free-field Lagrangian as well as the propagator for spin-$\frac{3}{2}$ particles contains an arbitrary parameter $A$. However, the $S$-matrix elements for the interaction of a spin-$\frac{3}{2}$ field with other fields are always independent of this parameter, provided the interaction Lagrangian is properly chosen. For a system of a spin-$\frac{3}{2}$ field coupled to a nucleon field and a pion field, a two-parameter ($A,Z$) interaction Lagrangian is introduced in such a way that it is invariant under a point transformation of the spin-$\frac{3}{2}$ field. The transition amplitude for such an interaction is independent of the parameter $A$. However, it does depend on the second parameter $Z$. By requiring the interaction to be consistent with the principles of second quantization, the value of the second parameter has been fixed. Therefore, the $\Delta \mathrm{}\left(1238\right)\mathrm{}$ contribution to the elastic $\pi N$ scattering amplitudes can be uniquely determined. Using this model for the evaluation of the $\Delta \mathrm{}\left(1238\right)\mathrm{}$ contribution and the chiral-invariant Lagrangians for the calculation of the nucleon pole and $\rho$-pole contributions, the $\pi N$ scattering lengths have been computed and compared with the experimental data. The calculated results are in good agreement with experiment.