If the Lagrangian density is a scalar and the 4-volume is a pseudo-scalar (w.r. to proper orthochronous LT), how is then action not a pseudo-scalar?

If it is a pseudo-scalar (i.e. the above reasoning is correct), why?

Related topics:

Is 4-volume element a scalar or a pseudoscalar in special relativity?

Deriving the action and the Lagrangian for a free massive point particle in Special Relativity

  • $\begingroup$ Is there any point at which the question of whether the action is a scalar or a pseudo-scalar is relevant for anything physically observable? $\endgroup$ – ACuriousMind Jun 12 '19 at 16:32

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