Abstract
In this work we review the derivation of Dirac and Weinberg equations based on a “principle of indistinguishability” for the (j,0) and (0,j) irreducible representations (irreps) of the homogeneous Lorentz group (HLG). We generalize this principle and explore its consequences for other irreps containing j≥1. We rederive Ahluwalia–Kirchbach equation using this principle and conclude that it yields \(\mathcal{O}(p^{2j} )\) equations of motion for any representation containing spin j and lower spins. We also use the obtained generators of the HLG for a given representation to explore the possibility of the existence of first order equations for that representation. We show that, except for j=\( - \frac{1}{2}\), there exists no Dirac-like equation for the (j,0)⊕(0,j) representation nor for the (\( - \frac{1}{2}\),\( - \frac{1}{2}\)) representation. We rederive Kemmer–Duffin–Petieau (KDP) equation for the (1,0)⊕(\( - \frac{1}{2}\),\( - \frac{1}{2}\))⊕(0,1) representation by this method and show that the (1,\( - \frac{1}{2}\))⊕(\( - \frac{1}{2}\),1) representation satisfies a Dirac-like equation which describes a multiplet of \(j = \frac{3}{2}{\text{ and }}j = \frac{1}{2}\) with masses m and m/2, respectively.
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Napsuciale, M. “Principle of Indistinguishability” and Equations of Motion for Particles with Spin. Foundations of Physics 33, 741–768 (2003). https://doi.org/10.1023/A:1025696823295
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DOI: https://doi.org/10.1023/A:1025696823295