Quantum-mechanical description of “particles” with complex spin

It is shown how the concept of the direct product $T^{(s_1)}\otimes T^{(s_2)}$ of two irreducible representations of the rotation group can be generalized to the case where the weights $s_1$ and $s_2$ of these irreducible representations are complex provided the difference $2s_1-2s_2$ is an integer; for complex $s_1$ and $s_2$, the representation $T^{(s_1)}\otimes T^{(s_2)}$ is defined as the restriction of the irreducible representation $T^{(s_1,s_2)}$ of the Lorenz group to the rotation group. This method does not offer an opportunity to consider a single irreducible representation $T^{(s)}$ for complex $s$. The definition of the representation $T^{(s_1)}\otimes T^{(s_2)}$ was used for the generalization of the concept of a spin wave function for a system of two particles with spins $s_1$ and $s_2$ ($2s_1-2s_2=n$) and for the generalization of the single- particle polarization density matrix to the case of complex spins. One cannot manage to introduce a wave function for an individual “particle” within the framework of this scheme; therefore such a “particle” is not observable or only observable to a limited extent. However, a system of two “particles” with complex spins $s_1$ and $s_2$ ($2s_1-2s_2=n$) can be in a state with integral or half-integral values of the total spin $\vert s_1-s_2\vert, \vert s_1-s_2\vert+1,\dots$; such a state may correspond to real particles. The proposed scheme may find application in the theory of unstable particles and also in quark theory if, going along with I. S. Shapiro, one assigns complex spin to quarks.