Right sign of spin rotation operator

Abstract

For the fermion transformation in the space all books of quantum mechanics propose to use the unitary operator $\widehat{U}_{\vec n}(\varphi)=\exp{(-i\frac\varphi2(\widehat\sigma\cdot\vec n))}$, where $\varphi$ is angle of rotation around the axis $\vec{n}$. But this operator turns the spin in inverse direction presenting the rotation to the left. The error of defining of $\widehat{U}_{\vec n}(\varphi)$ action is caused because the spin supposed as simple vector which is independent from $\widehat\sigma$-operator a priori. In this work it is shown that each fermion marked by number $i$ has own Pauli-vector $\widehat\sigma_i$ and both of them change together. If we suppose the global $\widehat\sigma$-operator and using the Bloch Sphere approach define for all fermions the common quantization axis $z$ the spin transformation will be the same: the right hand rotation around the axis $\vec{n}$ is performed by the operator $\widehat{U}^+_{\vec n}(\varphi)=\exp{(+i\frac\varphi2(\widehat\sigma\cdot\vec n))}$.