# Can spin angular momentum be understood as orbital angular momentum in extra dimensions?

It is to my understanding that in Kaluza-Klein theories the mass of particles can be understood as linear momentum in the extra dimensions.

Let's consider in $\mathbb{R}^{1,3}\times{}B$ space-time a massless spinor $\require{cancel}$ \begin{equation*} \cancel{D}_{4+D}\Psi=0 \end{equation*} let's now make this split \begin{equation*} \Psi=\psi(x)\otimes\phi(y) \end{equation*} where $\psi(x)$ is a spinor in 4 dimensions, and $\phi(y)$ an spinor in the compact space. Making a similar split in the operators in the Dirac equation \begin{equation*} (\cancel{D}_4\otimes{}I+I\otimes\cancel{D}_B)\Psi=0 \end{equation*} and introducing $\Psi$ \begin{equation*} (\cancel{D}_4\otimes{}I+I\otimes{}\cancel{D}_B)\psi\otimes\phi=0 \end{equation*} \begin{equation*} (\cancel{D}_4\psi\otimes{}\phi+\psi\otimes\cancel{D}_B\phi)=0 \end{equation*} now, if we demand to $\phi$ to be a eigenvector of $\cancel{D}_B$ with eigenvalue $m$ \begin{equation*} (\cancel{D}_4\psi\otimes{}\phi+m\psi\otimes\phi)=0 \end{equation*} \begin{equation*} (\cancel{D}_4-m)\psi\otimes{}\phi=0 \end{equation*} \begin{equation*} (\cancel{D}_4-m)\psi=0 \end{equation*} and we get massive fermions in 4 dimensions. We see also that the Dirac operator in the compact space acts as a kind of mass operator.

Could analoguously spin angular momentum be obtained from something related to angular momentum in the extra dimensions?

• Is it true that mass is linear momentum? My recollection is that the tower states are effectively standing waves of decreasing wavelength around the extra dimensions, and the mass is simply the energy of the standing wave. I have a vague recollection that angular momentum round a compact extra dimension looks like a charge. – John Rennie Dec 18 '14 at 9:59
• consider fermions inspirehep.net/record/10244?ln=es in this paper it is argued that we can get massive fermions in four dimensions if the eigenvalue of the Dirac operator of the compact space (which heuristically I link with the momentum) is understood as mass – Yossarian Dec 18 '14 at 10:29
• Sadly that's behind a paywall ... – John Rennie Dec 18 '14 at 10:42
• @JohnRennie I'll edit the question to get the interesting part of the info – Yossarian Dec 18 '14 at 10:46
• @silvrfück apparently \slashed doesn't work with MathJax. Would \cancel work better for you (looks too thick to me, but it might be better than red \slashed text)? – Ruslan Dec 18 '14 at 11:09