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)

In fact, it's neither. 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 $\mu \nu =23,31,12$). Its continuum form - 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 actually the Hodge dual of $M_{0}$, which is rank 3. The same considerations also apply to the stress tensor. The integral gives Noether charges for momentum and energy and is rank 1, while the integrand (the stress tensor density) is actually a tensor density of rank 2, its Hodge dual being more properly called "the stress tensor" and is of rank 2.