# Does spin generate rotation in ordinary 3-dimensional space?

Angular momenta are the generators of rotations. The orbital angular momentum, denoted by $\textbf{L}=(L_1,L_2,L_3)$, generates rotations in three-dimensional space in the planes $xy$, $yz$, $zx$ respectively.

The Spin $\textbf{S}=(S_1,S_2,S_3)$ also represents angular momentum which is intrinsic in nature. Being an angular momentum, it is also a generator rotation. But do they also generate ordinary rotation in 3-dimensional space?

• The spin operators also generates rotations in the three regular dimensions, it is just that they are used to rotate spinors, rather than regular vectors.
– Ihle
Commented Oct 4, 2015 at 13:25
• @SRS I am also now considering a similar problem. For me, rotation and boost are related with SU(2) and SL(2) operations, which can both be regarded as operations on a spin or a qubit. Obviously they lead to invariant Minkovski metric. If we further generate the single qubit/spin operation to double qubit operations ($GL(4)$), we will get general GR metric tensor. So I am thinking how 2 qubit operations are related with spacetime structure.
– XXDD
Commented Jun 20, 2017 at 6:02

I) The main point is that when we apply Noether's theorem for a field theory, the total angular momentum Noether current

$$J^{\mu,\nu\lambda}~=~L^{\mu,\nu\lambda}+S^{\mu,\nu\lambda}$$

splits in an orbital angular momentum current $$L^{\mu,\nu\lambda}=x^{\nu}T^{\mu,\lambda}-(\nu\leftrightarrow \lambda)$$

and an internal spin angular momentum Noether current $$S^{\mu,\nu\lambda}~=~\frac{\partial {\cal L}}{\partial \Phi_{,\mu}} \Sigma^{\nu\lambda}\Phi,$$

where $\Sigma^{\nu\lambda}$ furnishes a Lorentz representation of the field $\Phi$ in the field target space. For further details, see e.g. Ref. 1.

II) In particular for a scalar field theory, there is no spin, $\Sigma^{\nu\lambda}=0$ is generators for the trivial representation, and the angular momentum is generated purely from infinitesimal variations in spacetime, cf. e.g. this Phys.SE post.

References:

1. A. Bandyopadhyay, Improvement of the Stress-Energy Tensor using Spacetime symmetries, PhD thesis (2001); Chapter 2.