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| {{for|the notation|Ricci calculus}}
| | I'm April and I live in Buronzo. <br>I'm interested in Business and Management, Collecting cards and Danish art. I like to travel and reading fantasy.<br><br>Also visit my site ... [http://www.latansastore.com Tas selempang wanita] |
| {{quantum field theory}}
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| In [[physics]], specifically in [[relativistic quantum mechanics]] and [[quantum field theory]], the '''Pauli–Lubanski pseudovector''' named after [[Wolfgang Pauli]] and [[Józef Lubański]]<ref>{{cite doi|10.1016/S0031-8914(42)90113-7|noedit}}; {{cite doi|10.1016/S0031-8914(42)90114-9|noedit}}
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| </ref> is an [[operator (physics)|operator]] defined from the momentum and angular momentum, used in the quantum-relativistic description of [[angular momentum]]. It is the generator of the
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| [[little group]] of the Lorentz group, that is the maximal subgroup (with four generators) leaving the eigenvalues of the four-momentum vector {{math|P<sub>μ</sub>}} invariant.<ref>{{cite journal |
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| title=On unitary representations of the inhomogeneous Lorentz group |
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| author=Wigner, Eugene |
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| journal=The Annals of Mathematics |
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| volume=40 |
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| number=1 |
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| pages=149–204 |
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| year=1939 |
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| doi=10.2307/1968551 }}</ref>
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| ==Definition==
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| It is usually denoted by {{math|W}} (or less often by {{math|S}}) and defined by:<ref>{{cite book|title=Quantum Field Theory|author=L.H. Ryder|publisher=Cambridge University Press|edition=2nd|isbn=0-52147-8146|year=1996|page=62|url=http://books.google.co.uk/books?id=nnuW_kVJ500C&pg=PA62&dq=pauli-lubanski+pseudovector&hl=en&sa=X&ei=Wl1uUd75NtCZ0QWOp4HwDw&ved=0CDsQ6AEwAQ#v=onepage&q=pauli-lubanski%20pseudovector&f=false}}</ref><ref>{{cite book|title=General Principles of Quantum Field Theory|author=N.N. Bogolubov|publisher=Springer|edition=2nd|isbn=0-7923-0540-X|year=1989|page=273|url=http://books.google.co.uk/books?id=7VLMj4AvvicC&pg=PA273&dq=pauli-lubanski+pseudovector&hl=en&sa=X&ei=LF9uUa7XNoLw0gX914GACA&ved=0CEEQ6AEwAg#v=onepage&q=pauli-lubanski%20pseudovector&f=false}}</ref><ref>{{cite book|isbn=1-13950-4320|author=T. Ohlsson|title=Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory|publisher=Cambridge University Press|year=2011|page=11|url=http://books.google.co.uk/books?id=hRavtAW5EFcC&pg=PA11&dq=pauli-lubanski+pseudovector&hl=en&sa=X&ei=LF9uUa7XNoLw0gX914GACA&ved=0CEYQ6AEwAw#v=onepage&q=pauli-lubanski%20pseudovector&f=false}}</ref>
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| {{Equation box 1
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| |indent =::
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| |equation = <math>W_{\mu} \stackrel{\mathrm{def}}{=} ~ \tfrac{1}{2}\varepsilon_{\mu \nu \rho \sigma} J^{\nu \rho} P^\sigma ,</math>
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| |cellpadding= 6
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| |border
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| |border colour = #0073CF
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| |bgcolor=#F9FFF7}}
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| where
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| :*<math>\varepsilon_{\mu \nu \rho \sigma}</math> is the [[four dimensional]] totally [[antisymmetric tensor|antisymmetric]] [[Levi-Civita symbol]]
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| :*<math>J^{\nu \rho}</math> is the [[relativistic angular momentum tensor]] operator
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| :*<math>P^{\sigma}</math> is the [[four-momentum]].
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| In the language of [[exterior algebra]], it can be written as the [[Hodge dual]] of a [[Multivector|trivector]],<ref>{{cite book |author=R. Penrose| title=[[The Road to Reality]]| publisher= Vintage books|pages=568| year=2005 | isbn=978-00994-40680}}</ref>
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| :<math>\mathbf{W} = {}^\star(\mathbf{J}\wedge\mathbf{p}).</math>
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| {{math|W<sub>μ</sub>}} evidently satisfies
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| :<math>P^{\mu}W_{\mu}=0,</math>
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| as well as the following [[commutator]] relations,
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| :<math>\left[P^{\mu},W^{\nu}\right]=0,</math>
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| :<math>\left[J^{\mu \nu},W^{\rho}\right]=i \left( g^{\rho \nu} W^{\mu} - g^{\rho \mu} W^{\nu}\right),</math>
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| Consequently,
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| :<math>\left[W_{\mu},W_{\nu}\right]=-i \epsilon_{\mu \nu \rho \sigma} W^{\rho} P^{\sigma}. </math>
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| The scalar {{math|W<sub>μ</sub>W<sup>μ</sup>}} is a Lorentz invariant operator, and commutes with the four-momentum, and can thus serve as a label for irreducible representations of the Lorentz group. That is, it can serve as the label for the spin, a feature of the spacetime structure of the representation, over and above the label ({{math|P<sub>μ</sub>P<sup>μ</sup>}}) for the mass of states in a representation.
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| ==Massive fields==
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| In [[quantum field theory]], in the case of a massive field, the [[Casimir invariant]] {{math|W<sub>μ</sub>W<sup>μ</sup>}} describes the total [[spin quantum number|spin]] of the particle, with [[eigenvalues]]
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| :<math>W^2=W_{\mu}W^{\mu}=-m^2 s(s+1),</math>
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| where {{math|s}} is the [[spin quantum number]] of the particle and {{math|m}} is its [[rest mass]].
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| It is straightforward to see this in the [[rest frame]] of the particle, where {{math|1={{vec|W}} = m{{vec|J}}}} and {{nowrap|{{math|1=W<sup>0</sup> = 0}}}}, so that the little group amounts to the rotation group,
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| :<math>W_{\mu}W^{\mu}=-m^2 \vec{J}\cdot\vec{J}.</math>
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| Since this is a [[Lorentz invariant]] quantity, it will be the same in all other [[Frame of reference|reference frame]]s. It is also customary to take {{math|W<sup>3</sup>}} to describe the spin projection along the third direction in the rest frame.
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| In moving frames, decomposing {{nowrap|{{math|1=W = (W<sub>0</sub>, {{vec|W}})}}}} into components {{math|(W<sub>1</sub>, W<sub>2</sub>, W<sub>3</sub>)}}, with {{math|W<sub>1</sub>}} and {{math|W<sub>2</sub>}} orthogonal to {{math|{{vec|P}}}}, and {{math|W<sub>3</sub>}} parallel to {{math|{{vec|P}}}}, the Pauli–Lubanski vector may be expressed in terms of the spin vector {{math|{{vec|S}}}} = {{math|(S<sub>1</sub>, S<sub>2</sub>, S<sub>3</sub>)}} (similarly decomposed) as
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| :<math>W_0 = P S_3, \qquad W_1 = m S_1, \qquad W_2 = m S_2, \qquad W_3 = \frac{E}{c^2} S_3,</math>
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| where
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| :<math>E^2 = P^2 c^2 + m^2 c^4</math>
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| is the [[energy–momentum relation]].
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| The transverse components {{math|W<sub>1</sub>, W<sub>2</sub>}}, along with {{math|S<sub>3</sub>}}, satisfy the following commutator relations (which apply generally, not just to non-zero mass representations),
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| :<math>[W_1, W_2] = \tfrac{ih}{2\pi} ((E/c^2)^2 - (P/c)^2) S_3, \qquad [W_2, S_3] = \tfrac{ih}{2\pi} W_1, \qquad [S_3, W_1] = \tfrac{ih}{2\pi} W_2.</math>
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| For particles with non-zero mass, and the fields associated with such particles,
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| :<math>[W_1, W_2] = \tfrac{ih}{2\pi} m^2 S_3.</math>
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| ==Massless fields==
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| In general, in the case of non-massive representations, two cases may be distinguished.
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| In the more general case, the components of {{math|{{vec|W}}}} transverse to {{math|{{vec|P}}}} may be non-zero, thus yielding the family of representations referred to as the ''cylindrical'' luxons, their identifying property being that the components of {{math|{{vec|W}}}} form a Lie subalgebra isomorphic to the 2-dimensional Euclidean group {{math|ISO(2)}}, with the longitudinal component of {{math|{{vec|W}}}} playing the role of the rotation generator, and the transverse components the role of translation generators. This amounts to a [[group contraction]] of {{math|SO(3)}}, and leads to what are known as the '''continuous spin''' representations. However, there are no known physical cases of fundamental particles or fields in this family.
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| In a special case, {{math|{{vec|W}}}} is parallel to {{math|{{vec|P}}}}; or equivalently {{nowrap|{{math|1={{vec|W}} × {{vec|P}} = {{vec|0}}}}}}. For non-zero {{math|{{vec|W}}}}, this constraint can only be consistently imposed for [[luxons]], since the commutator of the two transverse components of {{math|{{vec|W}}}} is proportional to {{math|m<sup>2</sup>}} {{math|{{vec|J}}·{{vec|P}}}}.
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| For this family, the [[Helicity (particle physics)|helicity]] operator
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| :<math>W^0/P,</math>
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| represents an invariant, where
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| :<math>W^0 = -\vec{J}\cdot\vec{P}.</math>
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| All particles that interact with the [[Weak Nuclear Force]], for instance, fall into this family, since the definition of weak nuclear charge ( weak [[isospin]]) involves helicity, which, by above, must be an invariant. The appearance of non-zero mass in such cases must then be explained by other means, such as the [[Higgs mechanism]].
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| Even after accounting for such mass-generating mechanisms, however, the [[photon]] (and therefore the electromagnetic field) continues to fall into this class, although the other mass eigenstates of the carriers of the [[electroweak force]] (the W particle and anti-particle and Z particle) acquire non-zero mass.
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| Neutrinos were formerly considered to fall into this class as well. However, through [[neutrino oscillation]]s, it is now known that at least two of the three mass eigenstates of the left-helicity neutrino and right-helicity anti-neutrino each must have non-zero mass.
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| ==See also==
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| *[[Center of mass (relativistic)]]
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| *[[Wigner's classification]]
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| *[[Angular momentum operator]]
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| *[[Casimir operator]]
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| *[[Chirality (physics)|Chirality]]
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| *[[Pseudovector]]
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| *[[Pseudotensor]]
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| *[[Induced representation]]
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| ==References==
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| ===Notes===
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| {{reflist}}
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| ===Other===
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| *{{cite book|author=Sergeĭ Mikhaĭlovich Troshin, Nikolaĭ Evgenʹevich Ti͡urin|year=1994|publisher=World Scientific|title=Spin phenomena in particle interactions|url=http://books.google.co.uk/books?id=AU2DV1hKpuoC&pg=PA9&dq=pauli-lubanski+pseudovector&hl=en&sa=X&ei=blxuUeOYGcPv0gXtsYCADg&redir_esc=y#v=onepage&q=pauli-lubanski%20pseudovector&f=false|isbn=9-81021-6920}}
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| {{DEFAULTSORT:Pauli-Lubanski pseudovector}}
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| [[Category:Quantum field theory]]
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| [[Category:Representation theory of Lie algebras]]
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I'm April and I live in Buronzo.
I'm interested in Business and Management, Collecting cards and Danish art. I like to travel and reading fantasy.
Also visit my site ... Tas selempang wanita