In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 and mathematical physics

Mathematical physics

Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...

, the Euclidean group

Euclidean group

In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space...

SE(d)

of direct isometries is generated by translation

Translation

Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. Whereas interpreting undoubtedly antedates writing, translation began only after the appearance of written literature; there exist partial translations of the Sumerian Epic of...

s and rotation

Rotation

A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

s. Its Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

 is written
.

Noether currents

The Noether currents for the translations make up the stress-energy tensor

Stress-energy tensor

The stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...

 .

It satisfies the continuity equation

Continuity equation

A continuity equation in physics is a differential equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described...

.

And


gives the energy-momentum  at time t, which is time-independent.

The Noether current for a rotation about the point y is given by .

Because of the Lie algebra relations,
,

where 0 is the origin

Origin (mathematics)

In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...

.

And


gives the angular momentum

Angular momentum

In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

  at time t.

Definition

Define the spin tensor at a point to be the value of the Noether current at of a rotation about x,


Because of the continuity equation
,

we get


and therefore, the stress-energy tensor

Stress-energy tensor

The stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...

 isn't symmetric.

S gives the spin density and M gives the angular momentum density. The angular momentum is the sum of the orbital angular momentum and spin

Spin (physics)

In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

.

gives the torque

Torque

Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....

 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

Electromagnetic field

An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...

 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.