Quantization of an interacting spin-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mfrac><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac></mml:math>field and the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>Δ</mml:mi></mml:math>isobar

Quantization of the free and interacting Rarita-Schwinger field is considered using the Hamiltonian path-integral formulation. The particular interaction we study in detail is the $\ensuremath{\pi}N\ensuremath{\Delta}$ coupling used in the phenomenology of the pion-nucleon and nucleon-nucleon systems. Within the Dirac constraint analysis, we show that there is an excess of degrees of freedom in the model, as well as the inconsistency related to the Johnson-Sudarshan-Velo-Zwanzinger problem. It is further suggested that couplings invariant under the gauge transformation of the Rarita-Schwinger field are generally free from these inconsistencies. We then construct and briefly analyze some lowest in derivatives gauge-invariant $\ensuremath{\pi}N\ensuremath{\Delta}$ couplings.

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