Right sign of spin rotation operator

TL;DR

This paper clarifies the correct sign convention for the spin rotation operator in quantum mechanics, emphasizing the importance of the Pauli-vector's role and proposing a consistent formulation for fermion spin transformations.

Contribution

It demonstrates that the sign of the spin rotation operator should be positive for right-handed rotation when using the global Pauli-vector and Bloch Sphere approach.

Findings

Correct sign ensures proper spin rotation direction.

Aligns the operator with the physical rotation convention.

Provides a consistent framework for fermion spin transformations.

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 $\widehat{U}^+_{\vec n}(\varphi)=\exp{(+i\frac\varphi2(\widehat\sigma\cdot\vec n))}$.