[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/pauli-lubanski-pseudovector-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/pauli-lubanski-pseudovector-wikipedia\/","headline":"Pauli\u2013Lubanski pseudovector – Wikipedia","name":"Pauli\u2013Lubanski pseudovector – Wikipedia","description":"Operator in quantum field theory In physics, the Pauli\u2013Lubanski pseudovector is an operator defined from the momentum and angular momentum,","datePublished":"2016-11-07","dateModified":"2016-11-07","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/09c25c3c1d76a6da96c792cc45541e34b7004c79","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/09c25c3c1d76a6da96c792cc45541e34b7004c79","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/pauli-lubanski-pseudovector-wikipedia\/","wordCount":9447,"articleBody":"Operator in quantum field theoryIn physics, the Pauli\u2013Lubanski pseudovector is an operator defined from the momentum and angular momentum, used in the quantum-relativistic description of angular momentum. It is named after Wolfgang Pauli and J\u00f3zef Luba\u0144ski,[1]It describes the spin states of moving particles.[2] It is the generator of the little group of the Poincar\u00e9 group, that is the maximal subgroup (with four generators) leaving the eigenvalues of the four-momentum vector P\u03bc invariant.[3]Table of ContentsDefinition[edit]Little group[edit]Massive fields[edit]Massless fields[edit]Continuous spin representations[edit]Helicity representations[edit]See also[edit]References[edit]Definition[edit]It is usually denoted by W (or less often by S) and defined by:[4][5][6]W\u03bc=def12\u03b5\u03bc\u03bd\u03c1\u03c3J\u03bd\u03c1P\u03c3,{displaystyle W_{mu }mathrel {stackrel {text{def}}{=}} {tfrac {1}{2}}varepsilon _{mu nu rho sigma }J^{nu rho }P^{sigma },}whereIn the language of exterior algebra, it can be written as the Hodge dual of a trivector,[7]W=\u22c6(J\u2227p).{displaystyle mathbf {W} =star (mathbf {J} wedge mathbf {p} ).}Note W0=J\u2192\u22c5P\u2192{displaystyle W_{0}={vec {J}}cdot {vec {P}}}, and W\u2192=EJ\u2192\u2212P\u2192\u00d7K\u2192.{displaystyle {vec {W}}=E{vec {J}}-{vec {P}}times {vec {K}}.}W\u03bc evidently satisfiesP\u03bcW\u03bc=0,{displaystyle P^{mu }W_{mu }=0,}as well as the following commutator relations,[P\u03bc,W\u03bd]=0,[J\u03bc\u03bd,W\u03c1]=i(g\u03c1\u03bdW\u03bc\u2212g\u03c1\u03bcW\u03bd),{displaystyle {begin{aligned}left[P^{mu },W^{nu }right]&=0,\\left[J^{mu nu },W^{rho }right]&=ileft(g^{rho nu }W^{mu }-g^{rho mu }W^{nu }right),end{aligned}}}Consequently,[W\u03bc,W\u03bd]=\u2212i\u03f5\u03bc\u03bd\u03c1\u03c3W\u03c1P\u03c3.{displaystyle left[W_{mu },W_{nu }right]=-iepsilon _{mu nu rho sigma }W^{rho }P^{sigma }.}The scalar W\u03bcW\u03bc is a Lorentz-invariant operator, and commutes with the four-momentum, and can thus serve as a label for irreducible unitary representations of the Poincar\u00e9 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 relativistically invariant label P\u03bcP\u03bc for the mass of all states in a representation.Little group[edit]On an eigenspace S{displaystyle S} of the 4-momentum operator P{displaystyle P} with 4-momentum eigenvalue k{displaystyle k} of the Hilbert space of a quantum system (or for that matter the standard representation with \u211d4 interpreted as momentum space acted on by 5\u00d75 matrices with the upper left 4\u00d74 block an ordinary Lorentz transformation, the last column reserved for translations and the action effected on elements p{displaystyle p} (column vectors) of momentum space with 1 appended as a fifth row, see standard texts[8][9]) the following holds:[10]The irreducible unitary representation of the Poincar\u00e9 group are characterized by the eigenvalues of the two Casimir operators P2{displaystyle P^{2}} and W2{displaystyle W^{2}}. The best way to see that an irreducible unitary representation actually is obtained is to exhibit its action on an element with arbitrary 4-momentum eigenvalue p{displaystyle p} in the representation space thus obtained.[11]:\u200a62\u201374\u200aIrreducibility follows from the construction of the representation space.Massive fields[edit]In quantum field theory, in the case of a massive field, the Casimir invariant W\u03bcW\u03bc describes the total spin of the particle, with eigenvaluesW2=W\u03bcW\u03bc=\u2212m2s(s+1),{displaystyle W^{2}=W_{mu }W^{mu }=-m^{2}s(s+1),}where s is the spin quantum number of the particle and m is its rest mass.It is straightforward to see this in the rest frame of the particle, the above commutator acting on the particle’s state amounts to [Wj , Wk] = i \u03b5jkl Wl m; hence W\u2192 = mJ\u2192 and W0 = 0, so that the little group amounts to the rotation group,W\u03bcW\u03bc=\u2212m2J\u2192\u22c5J\u2192.{displaystyle W_{mu }W^{mu }=-m^{2}{vec {J}}cdot {vec {J}}.}Since this is a Lorentz invariant quantity, it will be the same in all other reference frames.It is also customary to take W3 to describe the spin projection along the third direction in the rest frame.In moving frames, decomposing W = (W0, W\u2192) into components (W1, W2, W3), with W1 and W2 orthogonal to P\u2192, and W3 parallel to P\u2192, the Pauli\u2013Lubanski vector may be expressed in terms of the spin vector S\u2192 = (S1, S2, S3) (similarly decomposed) asW0=PS3,W1=mS1,W2=mS2,W3=Ec2S3,{displaystyle {begin{aligned}W_{0}&=PS_{3},&W_{1}&=mS_{1},&W_{2}&=mS_{2},&W_{3}&={frac {E}{c^{2}}}S_{3},end{aligned}}}whereE2=P2c2+m2c4{displaystyle E^{2}=P^{2}c^{2}+m^{2}c^{4}}is the energy\u2013momentum relation.The transverse components W1, W2, along with S3, satisfy the following commutator relations (which apply generally, not just to non-zero mass representations),[W1,W2]=ih2\u03c0((Ec2)2\u2212(Pc)2)S3,[W2,S3]=ih2\u03c0W1,[S3,W1]=ih2\u03c0W2.{displaystyle {begin{aligned}[][W_{1},W_{2}]&={frac {ih}{2pi }}left(left({frac {E}{c^{2}}}right)^{2}-left({frac {P}{c}}right)^{2}right)S_{3},&[W_{2},S_{3}]&={frac {ih}{2pi }}W_{1},&[S_{3},W_{1}]&={frac {ih}{2pi }}W_{2}.end{aligned}}}For particles with non-zero mass, and the fields associated with such particles,[W1,W2]=ih2\u03c0m2S3.{displaystyle [W_{1},W_{2}]={frac {ih}{2pi }}m^{2}S_{3}.}Massless fields[edit]In general, in the case of non-massive representations, two cases may be distinguished.For massless particles, [11]:\u200a71\u201372\u200aW2=W\u03bcW\u03bc=\u2212E2((K2\u2212J1)2+(K1+J2)2)=def\u2212E2(A2+B2),{displaystyle W^{2}=W_{mu }W^{mu }=-E^{2}left((K_{2}-J_{1})^{2}+(K_{1}+J_{2})^{2}right)mathrel {stackrel {mathrm {def} }{=}} -E^{2}left(A^{2}+B^{2}right),}where K\u2192 is the dynamic mass moment vector. So, mathematically, P2 = 0 does not imply W2 = 0.Continuous spin representations[edit]In the more general case, the components of W\u2192 transverse to P\u2192 may be non-zero, thus yielding the family of representations referred to as the cylindrical luxons (“luxon” is another term for “massless particle”), their identifying property being that the components of W\u2192 form a Lie subalgebra isomorphic to the 2-dimensional Euclidean group ISO(2), with the longitudinal component of W\u2192 playing the role of the rotation generator, and the transverse components the role of translation generators. This amounts to a group contraction of 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. It can be argued that continuous spin states possess an internal degree of freedom not seen in observed massless particles.[11]:\u200a69\u201374\u200aHelicity representations[edit]In a special case, W\u2192{displaystyle {vec {W}}} is parallel to P\u2192;{displaystyle {vec {P}};} or equivalently W\u2192\u00d7P\u2192=0\u2192.{displaystyle {vec {W}}times {vec {P}}={vec {boldsymbol {0}}}.}  For non-zero W\u2192{displaystyle {vec {W}}} this constraint can only be consistently imposed for luxons (massless particles), since the commutator of the two transverse components of W\u2192{displaystyle {vec {W}}} is proportional to m2J\u2192\u22c5P\u2192.{displaystyle m^{2}{vec {J}}cdot {vec {P}},.} For this family, W2=0{displaystyle W^{2}=0} and W\u03bc=\u03bbP\u03bc{displaystyle W^{mu }=lambda ,P^{mu }} the invariant is, instead given by(W0)2=(W3)2,{displaystyle left(W^{0}right)^{2}=left(W^{3}right)^{2},}whereW0=\u2212J\u2192\u22c5P\u2192,{displaystyle W^{0}=-{vec {J}}cdot {vec {P}},}so the invariant is represented by the helicity operatorW0\/P.{displaystyle W^{0}\/P.}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. 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\u00b1  boson and anti-boson and  Z0  boson) acquire non-zero mass.Neutrinos were formerly considered to fall into this class as well. However, because neutrinos have been observed to oscillate in flavour, it is now known that at least two of the three mass eigenstates of the left-helicity neutrinos and right-helicity anti-neutrinos each must have non-zero mass.See also[edit]^ Luba\u0144ski & 1942A, pp.\u00a0310\u2013324 harvnb error: no target: CITEREFLuba\u0144ski1942A (help), Luba\u0144ski & 1942B, pp.\u00a0325\u2013338 harvnb error: no target: CITEREFLuba\u0144ski1942B (help)^ Brown 1994, pp.\u00a0180\u2013181^ Wigner 1939, pp.\u00a0149\u2013204^ Ryder 1996, p.\u00a062^ Bogolyubov 1989, p.\u00a0273^ Ohlsson 2011, p.\u00a011^ Penrose 2005, p.\u00a0568^ Hall 2015, Formula 1.12.^ Rossmann 2002, Chapter 2.^ Tung 1985, Theorem 10.13, Chapter 10.^ a b c Weinberg, Steven (1995). The Quantum Theory of Fields. Vol.\u00a01. Cambridge University Press. ISBN\u00a0978-0521550017.References[edit]Bogolyubov, N.N. (1989). General Principles of Quantum Field Theory (2nd\u00a0ed.). Springer Verlag. ISBN\u00a00-7923-0540-X.Brown, L. S. (1994). Quantum Field Theory. Cambridge University Press. ISBN\u00a0978-0-521-46946-3.Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Entroduction, Graduate Texts in Mathematics, vol.\u00a0222 (2nd\u00a0ed.), Springer, doi:10.1007\/978-3-319-13467-3, ISBN\u00a0978-3319134666, ISSN\u00a00072-5285Luba\u0144ski, J. K. (1942A). “Sur la theorie des particules \u00e9l\u00e9mentaires de spin quelconque. I”. Physica (in French). 9 (3): 310\u2013324. Bibcode:1942Phy…..9..310L. doi:10.1016\/S0031-8914(42)90113-7.Lubanski, J. K. (1942B). “Sur la th\u00e9orie des particules \u00e9l\u00e9mentaires de spin quelconque. II”. Physica (in French). 9 (3): 325\u2013338. Bibcode:1942Phy…..9..325L. doi:10.1016\/S0031-8914(42)90114-9.Ohlsson, T. (2011). Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory. Cambridge University Press. ISBN\u00a0978-1-139-50432-4.Penrose, R. (2005). The Road to Reality. Vintage books. ISBN\u00a0978-0-09-944068-0.Rossmann, Wulf (2002), Lie Groups – An Introduction Through Linear Groups, Oxford Graduate Texts in Mathematics, Oxford Science Publications, ISBN\u00a00-19-859683-9Ryder, L.H. (1996). Quantum Field Theory (2nd\u00a0ed.). Cambridge University Press. ISBN\u00a00-521-47814-6.Tung, Wu-Ki (1985). Group Theory in Physics (1st\u00a0ed.). New Jersey\u00b7London\u00b7Singapore\u00b7Hong Kong: World Scientific. ISBN\u00a0978-9971966577.Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol.\u00a01, Cambridge: Cambridge University Press, ISBN\u00a00-521-55001-7Wigner, E. P. (1939). “On unitary representations of the inhomogeneous Lorentz group”. Annals of Mathematics. 40 (1): 149\u2013204. Bibcode:1939AnMat..40..149W. doi:10.2307\/1968551. JSTOR\u00a01968551. MR\u00a01503456. 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