Talk:Spin tensor

Rank 2 or 3?

In section "Noether currents", you indicate the angular momentum tensor once as tensor of rank 3, $M_{y}^{\alpha \mu \nu }$ or $M_{0}^{\alpha \mu \nu }$, and once as tensor of rank 2, $M_{0}^{\mu \nu }$ or $M^{\mu \nu }$. So, what is the rank now? --195.138.38.10 (talk) 17:01, 8 September 2011 (UTC)

The moment/angular momentum tensor $M^{\mu \nu }$, given as the integral itself, is a rank 2 tensor, and provides the Noether charges associated respectively with boosts (for $\mu \nu =01,02,03$) and spatial rotations (for $\mu \nu =23,31,12$). The integrand $M_{0}^{\alpha \mu \nu }$, which is the spin tensor itself - is technically a rank 3 tensor density (rather than a tensor) that provides the 3-current for $M^{\mu \nu }$. The tensor associated with this is rank 3.

It's customary to use German script for densities, so that the integrand would actually be denoted ${\mathfrak {M}}_{0}^{\alpha \mu \nu }$, while the corresponding tensor would be $M_{0}^{\alpha \mu \nu }$. The components of both coincide only for Cartesian (and Minkowski) coordinates, otherwise they differ by the factor which appears in the integral measure; e.g. the measure in Cartesian coordinates $dx\wedge dy\wedge dz$ becomes $r^{2}sin\,\theta \,dr\wedge d\theta \wedge d\phi$ in spherical coordinates, so that ${\mathfrak {M}}_{0}$ = $r^{2}sin\,\theta \,M_{0}$, when expressed in spherical coordinates. The confusion between tensors and densities is prevalent throughout the Physics literature and is a contributing factor to other mistakes further down the line (e.g. the frequent citing in the literature of the wrong power of c for the coupling coefficient in the Einstein field equation).