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The action is the integral of the Lagrangian over time, or the integral of the Lagrangian Density over both time and space.
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Extended SUSY and superspace
Such an action might describe a massless point-particle with spin $1$; see https://doi.org/10.1088/1742-6596/1208/1/012006. … Question: How do I construct such an action with off-shell SUSY using the superspace formalism? …