Spin tensor


In mathematics, mathematical physics, and theoretical physics, the spin tensor is a quantity used to describe the rotational motion of particles in spacetime. The tensor has application in
general relativity and special relativity, as well as quantum mechanics, relativistic quantum mechanics, and quantum field theory.
The Euclidean group SE of direct isometries is generated by translations and rotations. Its Lie algebra is written.
This article uses Cartesian coordinates and tensor index notation.

Background on Noether currents

The Noether current for translations in space is momentum, while the current for increments in time is energy. These two statements combine into one in spacetime: translations in spacetime, i.e. a displacement between two events, is generated by the four-momentum P. Conservation of four-momentum is given by the continuity equation:
where is the stress–energy tensor, and ∂ are partial derivatives that make up the four gradient. Integrating over space:
gives the four-momentum vector at time t.
The Noether current for a rotation about the point y is given by a tensor of 3rd order, denoted. Because of the Lie algebra relations
where the 0 subscript indicates the origin, the integral:
gives the angular momentum tensor at time t.

Definition

The spin tensor is defined at a point x to be the value of the Noether current at x of a rotation about x,
The continuity equation
implies:
and therefore, the stress–energy tensor is not a symmetric tensor.
The quantity S is the density of spin angular momentum, and M is the density of orbital angular momentum. The total angular momentum is always the sum of spin and orbital contributions.
The relation:
gives the torque density showing the rate of conversion between the orbital angular momentum and spin.

Examples

Examples of materials with a nonzero spin density are molecular fluids, the electromagnetic field and turbulent fluids. For molecular fluids, the individual molecules may be spinning. The electromagnetic field can have circularly polarized light. For turbulent fluids, we may arbitrarily make a distinction between long wavelength phenomena and short wavelength phenomena. A long wavelength vorticity may be converted via turbulence into tinier and tinier vortices transporting the angular momentum into smaller and smaller wavelengths while simultaneously reducing the vorticity. This can be approximated by the eddy viscosity.