[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/bargmann-wigner-equations-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/bargmann-wigner-equations-wikipedia\/","headline":"Bargmann\u2013Wigner equations – Wikipedia","name":"Bargmann\u2013Wigner equations – Wikipedia","description":"before-content-x4 Wave equation for arbitrary spin particles This article uses the Einstein summation convention for tensor\/spinor indices, and uses hats","datePublished":"2014-06-06","dateModified":"2014-06-06","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\/d29163d0442efeea410e74e61a48bb76978099ac","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d29163d0442efeea410e74e61a48bb76978099ac","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/bargmann-wigner-equations-wikipedia\/","wordCount":7587,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4Wave equation for arbitrary spin particlesThis article uses the Einstein summation convention for tensor\/spinor indices, and uses hats for quantum operators.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In relativistic quantum mechanics and quantum field theory, the Bargmann\u2013Wigner equations describe free particles with non-zero mass and arbitrary spin j, an integer for bosons (j = 1, 2, 3 …) or half-integer for fermions (j = 1\u20442, 3\u20442, 5\u20442 …). The solutions to the equations are wavefunctions, mathematically in the form of multi-component spinor fields.They are named after Valentine Bargmann and Eugene Wigner.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Table of ContentsHistory[edit]Statement of the equations[edit]Lorentz group structure[edit]Formulation in curved spacetime[edit]See also[edit]References[edit]Notes[edit]Further reading[edit]Books[edit]Selected papers[edit]External links[edit]History[edit]Paul Dirac first published the Dirac equation in 1928, and later (1936) extended it to particles of any half-integer spin before Fierz and Pauli subsequently found the same equations in 1939, and about a decade before Bargman, and Wigner.[1]Eugene Wigner wrote a paper in 1937 about unitary representations of the inhomogeneous Lorentz group, or the Poincar\u00e9 group.[2] Wigner notes Ettore Majorana and Dirac used infinitesimal operators applied to functions. Wigner classifies representations as irreducible, factorial, and unitary.In 1948 Valentine Bargmann and Wigner published the equations now named after them in a paper on a group theoretical discussion of relativistic wave equations.[3]Statement of the equations[edit]For a free particle of spin j without electric charge, the BW equations are a set of 2j coupled linear partial differential equations, each with a similar mathematical form to the Dirac equation. The full set of equations are[1][4][5]       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4(\u2212\u03b3\u03bcP^\u03bc+mc)\u03b11\u03b11\u2032\u03c8\u03b11\u2032\u03b12\u03b13\u22ef\u03b12j=0(\u2212\u03b3\u03bcP^\u03bc+mc)\u03b12\u03b12\u2032\u03c8\u03b11\u03b12\u2032\u03b13\u22ef\u03b12j=0\u22ee(\u2212\u03b3\u03bcP^\u03bc+mc)\u03b12j\u03b12j\u2032\u03c8\u03b11\u03b12\u03b13\u22ef\u03b12j\u2032=0{displaystyle {begin{aligned}&left(-gamma ^{mu }{hat {P}}_{mu }+mcright)_{alpha _{1}alpha _{1}’}psi _{alpha ‘_{1}alpha _{2}alpha _{3}cdots alpha _{2j}}=0\\&left(-gamma ^{mu }{hat {P}}_{mu }+mcright)_{alpha _{2}alpha _{2}’}psi _{alpha _{1}alpha ‘_{2}alpha _{3}cdots alpha _{2j}}=0\\&qquad vdots \\&left(-gamma ^{mu }{hat {P}}_{mu }+mcright)_{alpha _{2j}alpha ‘_{2j}}psi _{alpha _{1}alpha _{2}alpha _{3}cdots alpha ‘_{2j}}=0\\end{aligned}}}which follow the pattern;(\u2212\u03b3\u03bcP^\u03bc+mc)\u03b1r\u03b1r\u2032\u03c8\u03b11\u22ef\u03b1r\u2032\u22ef\u03b12j=0{displaystyle left(-gamma ^{mu }{hat {P}}_{mu }+mcright)_{alpha _{r}alpha ‘_{r}}psi _{alpha _{1}cdots alpha ‘_{r}cdots alpha _{2j}}=0}(1)for r = 1, 2, … 2j. (Some authors e.g. Loide and Saar[4] use n = 2j to remove factors of 2. Also the spin quantum number is usually denoted by s in quantum mechanics, however in this context j is more typical in the literature). The entire wavefunction \u03c8 = \u03c8(r, t) has components\u03c8\u03b11\u03b12\u03b13\u22ef\u03b12j(r,t){displaystyle psi _{alpha _{1}alpha _{2}alpha _{3}cdots alpha _{2j}}(mathbf {r} ,t)}and is a rank-2j 4-component spinor field. Each index takes the values 1, 2, 3, or 4, so there are 42j components of the entire spinor field \u03c8, although a completely symmetric wavefunction reduces the number of independent components to 2(2j + 1). Further, \u03b3\u03bc = (\u03b30, \u03b3) are the gamma matrices, andP^\u03bc=i\u210f\u2202\u03bc{displaystyle {hat {P}}_{mu }=ihbar partial _{mu }}is the 4-momentum operator.The operator constituting each equation, (\u2212\u03b3\u03bcP\u03bc + mc) = (\u2212i\u0127\u03b3\u03bc\u2202\u03bc + mc), is a 4 \u00d7 4 matrix, because of the \u03b3\u03bc matrices, and the mc term scalar-multiplies the 4 \u00d7 4 identity matrix (usually not written for simplicity). Explicitly, in the Dirac representation of the gamma matrices:[1]\u2212\u03b3\u03bcP^\u03bc+mc=\u2212\u03b30E^c\u2212\u03b3\u22c5(\u2212p^)+mc=\u2212(I200\u2212I2)E^c+(0\u03c3\u22c5p^\u2212\u03c3\u22c5p^0)+(I200I2)mc=(\u2212E^c+mc0p^zp^x\u2212ip^y0\u2212E^c+mcp^x+ip^y\u2212p^z\u2212p^z\u2212(p^x\u2212ip^y)E^c+mc0\u2212(p^x+ip^y)p^z0E^c+mc){displaystyle {begin{aligned}-gamma ^{mu }{hat {P}}_{mu }+mc&=-gamma ^{0}{frac {hat {E}}{c}}-{boldsymbol {gamma }}cdot (-{hat {mathbf {p} }})+mc\\[6pt]&=-{begin{pmatrix}I_{2}&0\\0&-I_{2}\\end{pmatrix}}{frac {hat {E}}{c}}+{begin{pmatrix}0&{boldsymbol {sigma }}cdot {hat {mathbf {p} }}\\-{boldsymbol {sigma }}cdot {hat {mathbf {p} }}&0\\end{pmatrix}}+{begin{pmatrix}I_{2}&0\\0&I_{2}\\end{pmatrix}}mc\\[8pt]&={begin{pmatrix}-{frac {hat {E}}{c}}+mc&0&{hat {p}}_{z}&{hat {p}}_{x}-i{hat {p}}_{y}\\0&-{frac {hat {E}}{c}}+mc&{hat {p}}_{x}+i{hat {p}}_{y}&-{hat {p}}_{z}\\-{hat {p}}_{z}&-({hat {p}}_{x}-i{hat {p}}_{y})&{frac {hat {E}}{c}}+mc&0\\-({hat {p}}_{x}+i{hat {p}}_{y})&{hat {p}}_{z}&0&{frac {hat {E}}{c}}+mc\\end{pmatrix}}\\end{aligned}}}where \u03c3 = (\u03c31, \u03c32, \u03c33) = (\u03c3x, \u03c3y, \u03c3z) is a vector of the Pauli matrices, E is the energy operator, p = (p1, p2, p3) = (px, py, pz) is the 3-momentum operator, I2 denotes the 2 \u00d7 2 identity matrix, the zeros (in the second line) are actually 2 \u00d7 2 blocks of zero matrices.The above matrix operator contracts with one bispinor index of \u03c8 at a time (see matrix multiplication), so some properties of the Dirac equation also apply to the BW equations:E2=(pc)2+(mc2)2{displaystyle E^{2}=(pc)^{2}+(mc^{2})^{2}}Unlike the Dirac equation, which can incorporate the electromagnetic field via minimal coupling, the B\u2013W formalism comprises intrinsic contradictions and difficulties when the electromagnetic field interaction is incorporated. In other words, it is not possible to make the change P\u03bc \u2192 P\u03bc \u2212 eA\u03bc, where e is the electric charge of the particle and A\u03bc = (A0, A) is the electromagnetic four-potential.[6][7] An indirect approach to investigate electromagnetic influences of the particle is to derive the electromagnetic four-currents and multipole moments for the particle, rather than include the interactions in the wave equations themselves.[8][9]Lorentz group structure[edit]The representation of the Lorentz group for the BW equations is[6]DBW=\u2a02r=12j[Dr(1\/2,0)\u2295Dr(0,1\/2)].{displaystyle D^{mathrm {BW} }=bigotimes _{r=1}^{2j}left[D_{r}^{(1\/2,0)}oplus D_{r}^{(0,1\/2)}right],.}where each Dr is an irreducible representation. This representation does not have definite spin unless j equals 1\/2 or 0. One may perform a Clebsch\u2013Gordan decomposition to find the irreducible (A, B) terms and hence the spin content. This redundancy necessitates that a particle of definite spin j that transforms under the DBW representation satisfies field equations.The representations D(j, 0) and D(0, j) can each separately represent particles of spin j. A state or quantum field in such a representation would satisfy no field equation except the Klein\u2013Gordon equation.Formulation in curved spacetime[edit]Following M. Kenmoku,[10] in local Minkowski space, the gamma matrices satisfy the anticommutation relations:[\u03b3i,\u03b3j]+=2\u03b7ijI4{displaystyle [gamma ^{i},gamma ^{j}]_{+}=2eta ^{ij}I_{4}}where \u03b7ij = diag(\u22121, 1, 1, 1) is the Minkowski metric. For the Latin indices here, i, j = 0, 1, 2, 3. In curved spacetime they are similar:[\u03b3\u03bc,\u03b3\u03bd]+=2g\u03bc\u03bd{displaystyle [gamma ^{mu },gamma ^{nu }]_{+}=2g^{mu nu }}where the spatial gamma matrices are contracted with the vierbein bi\u03bc to obtain \u03b3\u03bc = bi\u03bc \u03b3i, and g\u03bc\u03bd = bi\u03bcbi\u03bd is the metric tensor. For the Greek indices; \u03bc, \u03bd = 0, 1, 2, 3.A covariant derivative for spinors is given byD\u03bc=\u2202\u03bc+\u03a9\u03bc{displaystyle {mathcal {D}}_{mu }=partial _{mu }+Omega _{mu }}with the connection \u03a9 given in terms of the spin connection \u03c9 by:\u03a9\u03bc=14\u2202\u03bc\u03c9ij(\u03b3i\u03b3j\u2212\u03b3j\u03b3i){displaystyle Omega _{mu }={frac {1}{4}}partial _{mu }omega ^{ij}(gamma _{i}gamma _{j}-gamma _{j}gamma _{i})}The covariant derivative transforms like \u03c8:D\u03bc\u03c8\u2192D(\u039b)D\u03bc\u03c8{displaystyle {mathcal {D}}_{mu }psi rightarrow D(Lambda ){mathcal {D}}_{mu }psi }With this setup, equation (1) becomes:(\u2212i\u210f\u03b3\u03bcD\u03bc+mc)\u03b11\u03b11\u2032\u03c8\u03b11\u2032\u03b12\u03b13\u22ef\u03b12j=0(\u2212i\u210f\u03b3\u03bcD\u03bc+mc)\u03b12\u03b12\u2032\u03c8\u03b11\u03b12\u2032\u03b13\u22ef\u03b12j=0\u22ee(\u2212i\u210f\u03b3\u03bcD\u03bc+mc)\u03b12j\u03b12j\u2032\u03c8\u03b11\u03b12\u03b13\u22ef\u03b12j\u2032=0.{displaystyle {begin{aligned}&(-ihbar gamma ^{mu }{mathcal {D}}_{mu }+mc)_{alpha _{1}alpha _{1}’}psi _{alpha ‘_{1}alpha _{2}alpha _{3}cdots alpha _{2j}}=0\\&(-ihbar gamma ^{mu }{mathcal {D}}_{mu }+mc)_{alpha _{2}alpha _{2}’}psi _{alpha _{1}alpha ‘_{2}alpha _{3}cdots alpha _{2j}}=0\\&qquad vdots \\&(-ihbar gamma ^{mu }{mathcal {D}}_{mu }+mc)_{alpha _{2j}alpha ‘_{2j}}psi _{alpha _{1}alpha _{2}alpha _{3}cdots alpha ‘_{2j}}=0,.\\end{aligned}}}See also[edit]References[edit]Notes[edit]Further reading[edit]Books[edit]Selected papers[edit]External links[edit]Relativistic wave equations:Dirac matrices in higher dimensions, Wolfram Demonstrations ProjectLearning about spin-1 fields, P. Cahill, K. Cahill, University of New Mexico[permanent dead link]Field equations for massless bosons from a Dirac\u2013Weinberg formalism, R.W. Davies, K.T.R. Davies, P. Zory, D.S. Nydick, American Journal of PhysicsQuantum field theory I, Martin Moj\u017ei\u0161 Archived 2016-03-03 at the Wayback MachineThe Bargmann\u2013Wigner Equation: Field equation for arbitrary spin, FarzadQassemi, IPM School and Workshop on Cosmology, IPM, Tehran, IranLorentz groups in relativistic quantum physics:Representations of Lorentz Group, indiana.eduAppendix C: Lorentz group and the Dirac algebra, mcgill.ca[permanent dead link]The Lorentz Group, Relativistic Particles, and Quantum Mechanics, D. E. Soper, University of Oregon, 2011Representations of Lorentz and Poincar\u00e9 groups, J. Maciejko, Stanford UniversityRepresentations of the Symmetry Group of Spacetime, K. Drake, M. Feinberg, D. Guild, E. Turetsky, 2009     (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/bargmann-wigner-equations-wikipedia\/#breadcrumbitem","name":"Bargmann\u2013Wigner equations – Wikipedia"}}]}]