I was following the book 'Quantum Field theory and the Standard model' written by Mattehw D. Schwartz and I found something unconvincing in p.174 of the book.

In that page, it argues that the zeroth component of the Noether current (to the Dirac Lagrangian) $J\_{\mu}=\bar{\psi}\gamma\_{\mu}\psi$ corresponding to the global symmetry of the spinors $\psi\rightarrow e^{-i\alpha}\psi$ should be interpreted as the probability density of the fermions. That is, it argues that $J\_{0}=\psi^{\dagger}\psi=\psi\_{L}^{\dagger}\psi\_{L}+ \psi\_{R}^{\dagger}\psi\_{R}$ is the probability density of the Fermions.

My issue here is, although this $J\_{0}$ is a quantity that is conserved according to the continuity equation for the Noether current, it evidently is 'NOT' Lorentz invariant. This in turn means that the probability density should be different depending on what reference frame we choose to be in. How can such a quantity which is not Lorentz invariant be physically interpreted as probability density? My guess is that the 'GLOBAL' amount of charge (or probability) can somehow be Lorentz invariant when we use our usual reasoning that the boundary values should vanish at infinity, but isn't it the 'LOCAL' probability that has to be Lorentz invariant in order to constitute a physically meaningful theory? Probability of an event of observing a particle at a particular spacetime grid has to be reference frame independent at least from my understanding.