Inmathematics,mathematical physics, andtheoretical physics, thespin tensor is a quantity used to describe therotational motion of particles inspacetime. The spin tensor has application ingeneral relativity andspecial relativity, as well asquantum mechanics,relativistic quantum mechanics, andquantum field theory.

Thespecial Euclidean group SE(d) ofdirect isometries is generated bytranslations androtations. ItsLie algebra is written${\displaystyle \mathfrak{s}\mathfrak{e}(d)}$.

This article usesCartesian coordinates andtensor index notation.

TheNoether 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-momentumP. Conservation of four-momentum is given by thecontinuity equation:

$${\displaystyle {\mathrm{\partial }}_{\nu }{T}^{\mu \nu }=0,}$$

where${\displaystyle T^{\mu \nu }}$ is thestress–energy tensor, and ∂ arepartial derivatives that make up thefour-gradient (in non-Cartesian coordinates this must be replaced by thecovariant derivative). Integrating over space:

$${\displaystyle \int d^{3}x{T}^{\mu 0}\left(\overrightarrow{x},t\right)=P^{\mu }}$$

gives thefour-momentum vector at timet.

The Noether current for a rotation about the pointy is given by a tensor of 3rd order, denoted${\displaystyle M_y^{\alpha \beta \mu }}$. Because of theLie algebra relations

$${\displaystyle M_y^{\alpha \beta \mu }(x)=M_0^{\alpha \beta \mu }(x)+y^{\alpha }{T}^{\beta \mu }(x)-y^{\beta }{T}^{\alpha \mu }(x),}$$

where the 0 subscript indicates theorigin (unlike momentum, angular momentum depends on the origin), the integral:

$${\displaystyle \int d^{3}x{M}_0^{\mu \nu }(\overrightarrow{x},t)}$$

gives theangular momentum tensor${\displaystyle M^{\mu \nu }}$ at timet.

Thespin tensor is defined at a pointx to be the value of the Noether current atx of a rotation aboutx,

$${\displaystyle S^{\alpha \beta \mu }(\mathbf{x})\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{M}_x^{\alpha \beta \mu }(\mathbf{x})=M_0^{\alpha \beta \mu }(\mathbf{x})+x^{\alpha }{T}^{\beta \mu }(\mathbf{x})-x^{\beta }{T}^{\alpha \mu }(\mathbf{x})}$$

The continuity equation

$${\displaystyle {\mathrm{\partial }}_{\mu }{M}_0^{\alpha \beta \mu }=0,}$$

implies:

$${\displaystyle {\mathrm{\partial }}_{\mu }{S}^{\alpha \beta \mu }=T^{\beta \alpha }-T^{\alpha \beta }\ne 0}$$

and therefore, thestress–energy tensor is not asymmetric tensor.

The quantityS is the density ofspin angular momentum (spin in this case is not only for a point-like particle, but also for an extended body), andM is the density of orbital angular momentum. The total angular momentum is always the sum of spin and orbital contributions.

The relation:

$${\displaystyle T_{ij}-T_{ji}}$$

gives thetorque density showing the rate of conversion between the orbital angular momentum and spin.

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

• Belinfante–Rosenfeld stress–energy tensor
• Poincaré group
• Lorentz group
• Relativistic angular momentum
• Mathisson–Papapetrou–Dixon equations
• Pauli–Lubanski pseudovector

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• Tensors
• Special relativity
• General relativity
• Quantum mechanics
• Quantum field theory
• Lie groups

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