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5

When you write the Dirac equation in a curved spacetime, in the context of General Relativity (which allows curvature, but not torsion) , you have a spin connection : $$\nabla_\mu\psi=\left(\partial_{\mu}-\frac i4\omega_{\mu}^{IJ}\sigma_{IJ}\right)\psi$$ Now, the Einstein-Cartan theory is not General Relativity, because it allows curvature, but also ...

3

You don't need to integrate anything here. The wavefunction is represented as a two component spinor here, which represents the probability of observing a certain spin along some axis. The components of the spinor refer to how the spin of the particle is oriented, they do not represent $x$ and $y$ components of a vector. So the inner product defined for such ...

1

Spin eigenvectors are the same for the electron and the positron. The transition amplitude between the singlet state $s$, and for instance, a state up for electron and down for positron may be written (up to a complex unit phase) : \$A = ((up_1)^\dagger \otimes (down_2)^\dagger) s \\=((\chi_+(\theta_1, \phi_1))^\dagger \otimes \chi_-(\theta_2, ...

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