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| :''Not to be confused with [[tensor product]]s of [[spin representation]]s.''
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| 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
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| [[general relativity]] and [[special relativity]], as well as [[quantum mechanics]], [[relativistic quantum mechanics]], and [[quantum field theory]].
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| The [[Euclidean group]] SE(d) of [[Euclidean group#Direct and indirect isometries|direct isometries]] is generated by [[translation]]s and [[rotation]]s. Its [[Lie algebra]] is written <math>\mathfrak{se}(d)</math>.
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| This article uses [[Cartesian coordinates]] and [[tensor index notation]].
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| ==Background on Noether currents==
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| 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]]:
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| :<math>\partial_\nu T^{\mu\nu}=0\,,</math>
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| where <math>T^{\mu\nu} \,</math> is the [[stress-energy tensor]], and ∂ are [[partial derivative]]s that make up the [[four gradient]] (in non-Cartesian coordinates this must be replaced by the [[covariant derivative]]). Integrating over spacetime: | |
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| :<math>\int d^4 x T^{\mu 0}(\vec{x},t) = P^\mu</math>
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| gives the [[four-momentum]] vector at time ''t''.
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| The Noether current for a rotation about the point ''y'' is given by a tensor of 3rd order, denoted <math>M^{\alpha\beta\mu}_y</math>. Because of the [[Lie algebra]] relations
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| :<math>M^{\alpha\beta\mu}_y(x)=M^{\alpha\beta\mu}_0(x)+y^\alpha T^{\beta\mu}(x)-y^\beta T^{\alpha\mu}(x)\,,</math>
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| where the 0 subscript indicates the [[Origin (mathematics)|origin]] (unlike momentum, angular momentum depends on the origin), the integral:
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| :<math>\int d^4 x M^{\mu\nu}_0(\vec{x},t)</math>
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| gives the angular momentum tensor <math>M^{\mu\nu} \,</math> at time ''t''.
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| ==Definition==
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| The '''spin tensor''' is defined at a point '''x''' to be the value of the Noether current at '''x''' of a rotation about ''x'',
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| :<math>S^{\alpha\beta\mu}(\mathbf{x})\ \stackrel{\mathrm{def}}{=}\ M^{\alpha\beta\mu}_x(\mathbf{x})=M^{\alpha\beta\mu}_0(\mathbf{x})+x^\alpha T^{\beta\mu}(\mathbf{x})-x^\beta T^{\alpha\mu}(\mathbf{x})</math>
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| The continuity equation
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| :<math>\partial_\mu M^{\alpha\beta\mu}_0=0\,,</math>
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| implies:
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| :<math>\partial_\mu S^{\alpha\beta\mu}=T^{\beta\alpha}-T^{\alpha\beta} \neq 0</math>
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| and therefore, the [[stress-energy tensor]] is not a [[symmetric tensor]].
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| The quantity ''S'' gives the [[spin density]] and ''M'' gives the angular momentum density. The angular momentum is the sum of the [[angular momentum operator|orbital angular momentum]] and [[spin (physics)|spin]] (spin in this case is not only for a point-like particle, but also for an extended bodies).
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| The relation:
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| :<math> T_{ij} - T_{ji} </math>
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| gives the [[torque]] density showing the rate of conversion between the orbital angular momentum and spin.
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| ==Examples==
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| Examples of materials with a nonzero spin density are [[molecular fluid]]s, the [[electromagnetic field]] and [[turbulent fluid]]s. 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]].
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| ==See also==
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| * [[Poincaré group]]
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| * [[Lorentz group]]
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| * [[Relativistic angular momentum]]
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| * [[Center of mass (relativistic)]]
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| * [[Mathisson–Papapetrou–Dixon equations]]
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| * [[Pauli–Lubanski pseudovector]]
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| ==References==
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| *{{cite book|title=General Relativity, Astrophysics, and Cosmology|author=A. K. Raychaudhuri, S. Banerji, A. Banerjee|year=2003|pages=66–67|publisher=Springer|series=Astronomy and astrophysics library|isbn=038-740-628-X|url=http://books.google.co.uk/books?id=PfZRZU_a8EAC&pg=PA66&dq=spin+tensor+general+relativity&hl=en&sa=X&ei=kWJtUqnwNKTT0QWJ_ICACg&redir_esc=y#v=onepage&q=spin%20tensor%20general%20relativity&f=false}}
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| *{{cite book |pages=156–159, §5.11| author=J.A. Wheeler, C. Misner, K.S. Thorne| title=[[Gravitation (book)|Gravitation]]| publisher=W.H. Freeman & Co| year=1973 | isbn=0-7167-0344-0}}
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| *{{cite news|author=L. M. Butcher, A. Lasenby, M. Hobson|arxiv=1210.0831|journal=Phys. Rev. D |title=Localizing the Angular Momentum of Linear Gravity|url=http://arxiv.org/abs/1210.0831|year=2012|doi=10.1103/PhysRevD.86.084012}}
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| *{{cite news|author=T. Banks|title=Modern Quantum Field Theory: A Concise Introduction|url=http://books.google.co.uk/books?id=tn4YO21Xh0IC&pg=PT92&dq=spin+tensor+and+noether+currents&hl=en&sa=X&ei=rvl1UvSWEI-ThgeN3oHACw&redir_esc=y#v=onepage&q=spin%20tensor%20and%20noether%20currents&f=false
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| |year=2008|isbn=113-947-389-1|publisher=Cambridge University Press}}
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| *{{cite news|author=S. Kopeikin, M.Efroimsky, G. Kaplan|title=Relativistic Celestial Mechanics of the Solar System|publisher=John Wiley & Sons|year=2011|isbn=352-763-457-6|url=http://books.google.co.uk/books?id=uN5_DQWSR14C&pg=PT453&dq=spin+tensor+and+noether+currents&hl=en&sa=X&ei=rvl1UvSWEI-ThgeN3oHACw&redir_esc=y#v=onepage&q=spin%20tensor%20and%20noether%20currents&f=false}}
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| *{{cite news|title=A spinor approach to the Lanczos spin tensor|journal=II Nuovo Cimento A|series=10|year=1968|volume=57|issue=4|pages=638–648|author=W. F. Maher, J. D. Zund|publisher=Springer|url=http://link.springer.com/article/10.1007%2FBF02751371#page-1}}
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| ==External links==
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| *{{cite web|title=Canonical Formulation of Spin in General Relativity (Dissertation)|author=von Jan Steinhoff|accessdate=2013-10-27|url=http://jan-steinhoff.de/physics/phd/phd_steinhoff.pdf}}
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| {{tensor}}
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| {{DEFAULTSORT:Spin Tensor}}
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| [[Category:Tensors]]
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| [[Category:Special relativity]]
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| [[Category:General relativity]]
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| [[Category:Quantum mechanics]]
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| [[Category:Quantum field theory]]
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| [[Category:Lie groups]]
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Alyson is what my spouse enjoys to contact me but I don't like when individuals use my complete title. North Carolina is where we've been living for many years and will never transfer. He works as a bookkeeper. I am truly fond of handwriting but I can't make it my occupation really.
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