Dirac action momenta conjugate to conjugate field

Question

Consider the Dirac action $S=\int d^4x\bar{\psi}(x)(i\not\partial-m)\psi(x)$. Since there are no time derivative of $\bar{\psi}$, we get the constraint that its canonical momenta vanishes. This constraint is of course first class. Does this mean that the Dirac equation has a gauge symmetry? Why do we usually not care about it when doing the canonical quantization of the Dirac field? Thanks!

Answer 1

The are 2 primary constraints rather than 1. (Recall that there is also 1 primary constraint for the complex conjugate field). The 2 primary constraints do not (super)Poisson-commute, so they are second-class rather than first-class constraints. And therefore no gauge symmetry. See e.g. my related Phys.SE answer here.