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Bargmann–Wigner equations

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Wave equation for arbitrary spin particles

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Inrelativistic quantum mechanics andquantum field theory, theBargmann–Wigner equations describefree particles with non-zero mass and arbitraryspinj, an integer forbosons (j = 1, 2, 3 ...) or half-integer forfermions (j =1⁄2,3⁄2,5⁄2 ...). The solutions to the equations arewavefunctions, mathematically in the form of multi-componentspinor fields.

They are named afterValentine Bargmann andEugene Wigner.

History

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Paul Dirac first published theDirac 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 aboutunitary representations of the inhomogeneousLorentz group, or thePoincaré group.^[2] Wigner notesEttore Majorana and Dirac used infinitesimal operators applied to functions. Wigner classifies representations as irreducible, factorial, and unitary.

In 1948Valentine 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

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For a free particle of spinj withoutelectric charge, the BW equations are a set of2j coupledlinear partial differential equations, each with a similar mathematical form to theDirac equation. The full set of equations are:^{[note 1]}^[1]^[4]^[5]

{\begin{aligned}&\left(-\gamma ^{\mu }{\hat {P}}_{\mu }+mc\right)_{\alpha _{1}\alpha _{1}'}\psi _{\alpha '_{1}\alpha _{2}\alpha _{3}\cdots \alpha _{2j}}=0\\&\left(-\gamma ^{\mu }{\hat {P}}_{\mu }+mc\right)_{\alpha _{2}\alpha _{2}'}\psi _{\alpha _{1}\alpha '_{2}\alpha _{3}\cdots \alpha _{2j}}=0\\&\qquad \vdots \\&\left(-\gamma ^{\mu }{\hat {P}}_{\mu }+mc\right)_{\alpha _{2j}\alpha '_{2j}}\psi _{\alpha _{1}\alpha _{2}\alpha _{3}\cdots \alpha '_{2j}}=0\\\end{aligned}}

which follow the pattern;

$\left(-\gamma ^{\mu }{\hat {P}}_{\mu }+mc\right)_{\alpha _{r}\alpha '_{r}}\psi _{\alpha _{1}\cdots \alpha '_{r}\cdots \alpha _{2j}}=0$

forr = 1, 2, ... 2j. (Some authors e.g. Loide and Saar^[4] usen = 2j to remove factors of 2. Also thespin quantum number is usually denoted bys in quantum mechanics, however in this contextj is more typical in the literature). The entire wavefunctionψ =ψ(r,t) has components

\psi _{\alpha _{1}\alpha _{2}\alpha _{3}\cdots \alpha _{2j}}(\mathbf {r} ,t)

and is a rank-2j 4-componentspinor field. Each index takes the values 1, 2, 3, or 4, so there are4^2j components of the entire spinor fieldψ, although a completely symmetric wavefunction reduces the number of independent components to2(2j + 1). Further,γ^μ = (γ⁰,γ) are thegamma matrices, and

{\hat {P}}_{\mu }=i\hbar \partial _{\mu }

is the4-momentum operator.

The operator constituting each equation,(−γ^μP_μ +mc) = (−iħγ^μ∂_μ +mc), is a4 × 4 matrix, because of theγ^μ matrices, and themc termscalar-multiplies the4 × 4identity matrix (usually not written for simplicity). Explicitly, in theDirac representation of the gamma matrices:^[1]

{\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}}

whereσ = (σ₁, σ₂, σ₃) = (σ_x, σ_y, σ_z) is a vector of thePauli matrices,E is theenergy operator,p = (p₁,p₂,p₃) = (p_x,p_y,p_z) is the3-momentum operator,I₂ denotes the2 × 2identity matrix, the zeros (in the second line) are actually2 × 2blocks ofzero matrices.

The above matrix operatorcontracts with one bispinor index ofψ at a time (seematrix multiplication), so some properties of the Dirac equation also apply to the BW equations:

the equations are Lorentz covariant,
all components of the solutionsψ also satisfy theKlein–Gordon equation, and hence fulfill the relativisticenergy–momentum relation,

E^{2}=(pc)^{2}+(mc^{2})^{2}

second quantization is still possible.

Unlike the Dirac equation, which can incorporate the electromagnetic field viaminimal coupling, the B–W formalism comprises intrinsic contradictions and difficulties when the electromagnetic field interaction is incorporated. In other words, it is not possible to make the changeP_μ →P_μ −eA_μ, wheree is theelectric charge of the particle andA_μ = (A₀,A) is theelectromagnetic four-potential.^[6]^[7] An indirect approach to investigate electromagnetic influences of the particle is to derive the electromagneticfour-currents andmultipole moments for the particle, rather than include the interactions in the wave equations themselves.^[8]^[9]

Lorentz group structure

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Therepresentation of the Lorentz group for the BW equations is^[6]

D^{\mathrm {BW} }=\bigotimes _{r=1}^{2j}\left[D_{r}^{(1/2,0)}\oplus D_{r}^{(0,1/2)}\right]\,.

where eachD_r is an irreducible representation. This representation does not have definite spin unlessj equals 1/2 or 0. One may perform aClebsch–Gordan decomposition to find the irreducible(A,B) terms and hence the spin content. This redundancy necessitates that a particle of definite spinj that transforms under theD^BW representation satisfies field equations.

The representationsD^{(j, 0)} andD^(0,j) can each separately represent particles of spinj. A state or quantum field in such a representation would satisfy no field equation except the Klein–Gordon equation.

Formulation in curved spacetime

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Main articles:spacetime andCartan formalism (physics)

Following M. Kenmoku,^[10] in local Minkowski space, the gamma matrices satisfy theanticommutation relations:

[\gamma ^{i},\gamma ^{j}]_{+}=2\eta ^{ij}I_{4}

whereη^ij = diag(−1, 1, 1, 1) is theMinkowski metric. For the Latin indices here,i, j = 0, 1, 2, 3. In curved spacetime they are similar:

[\gamma ^{\mu },\gamma ^{\nu }]_{+}=2g^{\mu \nu }

where the spatial gamma matrices are contracted with thevierbeinb_i^μ to obtainγ^μ =b_i^μ γⁱ, andg^μν =b^iμb_i^ν is themetric tensor. For the Greek indices;μ, ν = 0, 1, 2, 3.

Acovariant derivative for spinors is given by

{\mathcal {D}}_{\mu }=\partial _{\mu }+\Omega _{\mu }

with theconnectionΩ given in terms of thespin connectionω by:

\Omega _{\mu }={\frac {1}{4}}\partial _{\mu }\omega ^{ij}(\gamma _{i}\gamma _{j}-\gamma _{j}\gamma _{i})

The covariant derivative transforms likeψ:

{\mathcal {D}}_{\mu }\psi \rightarrow D(\Lambda ){\mathcal {D}}_{\mu }\psi

With this setup, equation (1) becomes:

{\begin{aligned}&(-i\hbar \gamma ^{\mu }{\mathcal {D}}_{\mu }+mc)_{\alpha _{1}\alpha _{1}'}\psi _{\alpha '_{1}\alpha _{2}\alpha _{3}\cdots \alpha _{2j}}=0\\&(-i\hbar \gamma ^{\mu }{\mathcal {D}}_{\mu }+mc)_{\alpha _{2}\alpha _{2}'}\psi _{\alpha _{1}\alpha '_{2}\alpha _{3}\cdots \alpha _{2j}}=0\\&\qquad \vdots \\&(-i\hbar \gamma ^{\mu }{\mathcal {D}}_{\mu }+mc)_{\alpha _{2j}\alpha '_{2j}}\psi _{\alpha _{1}\alpha _{2}\alpha _{3}\cdots \alpha '_{2j}}=0\,.\\\end{aligned}}

Notes

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^This article uses theEinstein summation convention fortensor/spinor indices, and useshats forquantum operators

References

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^^a ^b ^cE.A. Jeffery (1978)."Component Minimization of the Bargman–Wigner wavefunction".Australian Journal of Physics.31 (2): 137.Bibcode:1978AuJPh..31..137J.doi:10.1071/ph780137.
^E. Wigner (1937)."On Unitary Representations Of The Inhomogeneous Lorentz Group"(PDF).Annals of Mathematics.40 (1):149–204.Bibcode:1939AnMat..40..149W.doi:10.2307/1968551.JSTOR 1968551.S2CID 121773411. Archived fromthe original(PDF) on 2015-10-04. Retrieved2013-02-20.
^Bargmann, V.; Wigner, E. P. (1948)."Group theoretical discussion of relativistic wave equations".Proceedings of the National Academy of Sciences of the United States of America.34 (5):211–23.Bibcode:1948PNAS...34..211B.doi:10.1073/pnas.34.5.211.PMC 1079095.PMID 16578292.
^^a ^bR.K. Loide; I.Ots; R. Saar (2001). "Generalizations of the Dirac equation in covariant and Hamiltonian form".Journal of Physics A.34 (10):2031–2039.Bibcode:2001JPhA...34.2031L.doi:10.1088/0305-4470/34/10/307.
^H. Shi-Zhong; R. Tu-Nan; W. Ning; Z. Zhi-Peng (2002)."Wavefunctions for Particles with Arbitrary Spin".Communications in Theoretical Physics.37 (1): 63.Bibcode:2002CoTPh..37...63H.doi:10.1088/0253-6102/37/1/63.S2CID 123915995. Archived fromthe original on 2012-11-27. Retrieved2012-09-17.
^^a ^bT. Jaroszewicz; P.S. Kurzepa (1992). "Geometry of spacetime propagation of spinning particles".Annals of Physics.216 (2):226–267.Bibcode:1992AnPhy.216..226J.doi:10.1016/0003-4916(92)90176-M.
^C.R. Hagen (1970)."The Bargmann–Wigner method in Galilean relativity".Communications in Mathematical Physics.18 (2):97–108.Bibcode:1970CMaPh..18...97H.doi:10.1007/BF01646089.S2CID 121051722.
^Cédric Lorcé (2009). "Electromagnetic Properties for Arbitrary Spin Particles: Part 1 − Electromagnetic Current and Multipole Decomposition".arXiv:0901.4199 [hep-ph].
^Cédric Lorcé (2009). "Electromagnetic Properties for Arbitrary Spin Particles: Part 2 − Natural Moments and Transverse Charge Densities".Physical Review D.79 (11) 113011.arXiv:0901.4200.Bibcode:2009PhRvD..79k3011L.doi:10.1103/PhysRevD.79.113011.S2CID 17801598.
^K. Masakatsu (2012). "Superradiance Problem of Bosons and Fermions for Rotating Black Holes in Bargmann–Wigner Formulation".arXiv:1208.0644 [gr-qc].