# Anticommutation relations for Dirac field at non-equal times

I'm reading this paper by Alfredo Iorio and I have a doubt concerning the anticommutation relations he uses for the Dirac field.

Around eq. (2.25), he wants to find the unitary operator $$U$$ that implements quantum mechanically a Weyl transformation: $$\psi'(x)=U\psi(x)U^{-1}=e^{d_\psi\sigma(x)}\psi(x),~~\text{with }d_\psi=\frac{1-n}{2}.$$

He uses the anticommutation relations $$\{\psi(x),\psi^\dagger(y)\}=\delta^{(n)}(x-y)\tag{A}$$ $$\{\psi(x),\psi(y)\}=0\tag{B}$$ to find that the unitary operator is $$U=\exp\left\{-d_\psi\int d^ny~\sigma(y)\psi^\dagger(y)\psi(y)\right\}.$$

However, are those commutation relations correct? I thought that the correct ones are with equal time, i.e. $$\{\psi_\alpha(t,\mathbf{x}),\psi^\dagger_\beta(t,\mathbf{y})\}=\delta_{\alpha\beta}\delta^{(n-1)}(\mathbf{x}-\mathbf{y}).$$

Besides, I don't see that $$U$$ is unitary, that would require that $$\left(\psi^\dagger(y)\psi(y)\right)^\dagger=-\psi^\dagger(y)\psi(y)$$