
This article uses the Einstein summation convention for tensor/spinor indices, and uses hats for quantum operators.
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In relativistic quantum mechanics and quantum field theory, the Bargmann–Wigner equations (or BW equations or BWE) are relativistic wave equations which describe free particles of arbitrary spin j, an integer for bosons (j = 1, 2, 3 ...) or halfinteger for fermions (j = ^{1}⁄_{2}, ^{3}⁄_{2}, ^{5}⁄_{2} ...). The solutions to the equations are wavefunctions, mathematically in the form of multicomponent spinor fields. The spin quantum number is usually denoted by s in quantum mechanics, however in this context j is more typical in the literature (see references).
They were proposed by Valentine Bargmann and Eugene Wigner in 1948,^{[1]} using Lorentz group theory,^{[2]} and building on the work of those who pioneered quantum theory within the first half of the twentieth century.^{[3]}^{[4]}
Contents

Origin from the Dirac equation 1

BW equations 2

Induced matrices 3

Definition 3.1

Properties 3.2

Use in the BW formalism 3.3

Lorentz group structure 4

Lagrangian 5

Formulation in curved spacetime 6

See also 7

References 8

Notes 8.1

Further reading 8.2

Books 8.2.1

Selected papers 8.2.2

External links 9
Origin from the Dirac equation
For reference, the Dirac equation is summarized below. It is the basis for building relativistic wave equations with wavefunctions of higher spin.
The covariant form of the Dirac equation for an uncharged particle is:^{[5]}

(\gamma^\mu \hat{P}_\mu + mc)\Psi = 0 \,,


(1)

where Ψ = Ψ(r, t) is a rank1 4component spinor field, a function of the particle's position r and time t, with components ψ_{α} = ψ_{α}(r, t) in which α is a bispinor index that takes values 1, 2, 3, 4. Further, γ^{μ} = (γ^{0}, γ) are the gamma matrices, and

\hat{P}_\mu = i\hbar \partial_\mu
is the 4momentum operator. The operator constituting the entire equation, (−γ^{μ}P_{μ} + mc) = (−iħγ^{μ}∂_{μ} + mc), is a 4 × 4 matrix, because of the γ^{μ} matrices, and the mc term scalarmultiplies the 4 × 4 identity matrix (usually not written for simplicity). Explicitly, in the Dirac representation of the gamma matrices:^{[3]}

\begin{align} \gamma^\mu \hat{P}_\mu + mc & = \gamma^0 \frac{\hat{E}}{c}  \boldsymbol{\gamma}\cdot(\hat{\mathbf{p}}) + mc \\ & = \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 \\ & = \begin{pmatrix} \hat{E}/c+mc & 0 & \hat{p}_z & \hat{p}_x  i\hat{p}_y \\ 0 & \hat{E}/c+mc & \hat{p}_x + \hat{p}_y & \hat{p}_z \\ \hat{p}_z & (\hat{p}_x  i\hat{p}_y) & \hat{E}/c+mc & 0 \\ (\hat{p}_x + i\hat{p}_y) & \hat{p}_z & 0 & \hat{E}/c+mc \\ \end{pmatrix} \end{align}
where σ = (σ_{1}, σ_{2}, σ_{3}) = (σ_{x}, σ_{y}, σ_{z}) is a vector of the Pauli matrices, E is the energy operator, p = (p_{1}, p_{2}, p_{3}) = (p_{x}, p_{y}, p_{z}) is the 3momentum operator, I_{2} denotes the 2 × 2 identity matrix, the zeros (in the second line) are actually 2 × 2 blocks of zero matrices.
The Dirac equation (1) can be written as a coupled set of equations:

(\hat{E} + mc )\psi_{1,2} = (\boldsymbol{\sigma}\cdot\hat{\mathbf{p}})\psi_{3,4}


(1A)


(\hat{E} + mc )\psi_{3,4} = (\boldsymbol{\sigma}\cdot\hat{\mathbf{p}})\psi_{1,2}


(1B)

where

\Psi = \begin{pmatrix} \psi_{1,2} \\ \psi_{3,4} \\ \end{pmatrix}\,\quad \psi_{1,2} = \begin{pmatrix} \psi_1 \\ \psi_2 \\ \end{pmatrix}\,\quad \psi_{3,4} = \begin{pmatrix} \psi_3 \\ \psi_4 \\ \end{pmatrix}\,.
One 2component spinor ψ_{1,2} describes the spin1/2 fermion, the other ψ_{3,4} describes the antifermion.
For a charged particle moving in an electromagnetic field, minimal coupling can be introduced:

[\gamma^\mu (i\hbar \partial_\mu  eA_\mu)+mc]\Psi = 0


(1C)

where e is the electric charge of the particle and A_{μ} = (A_{0}, A) is the electromagnetic fourpotential.
BW equations
For a free particle of spin j, the BW equations are a set of 2j coupled linear partial differential equations, each with a similar mathematical form to the Dirac equation.
Uncharged massive particles
For a free particle with zero electric charge, the full set of equations are:^{[3]}^{[4]}^{[6]}

\begin{align} & (\gamma^\mu \hat{P}_\mu + mc)_{\alpha_1 \alpha_1'}\psi_{\alpha'_1 \alpha_2 \alpha_3 \cdots \alpha_{2j}} = 0 \\ & (\gamma^\mu \hat{P}_\mu + mc)_{\alpha_2 \alpha_2'}\psi_{\alpha_1 \alpha'_2 \alpha_3 \cdots \alpha_{2j}} = 0 \\ & \qquad \vdots \\ & (\gamma^\mu \hat{P}_\mu + mc)_{\alpha_{2j} \alpha'_{2j}}\psi_{\alpha_1 \alpha_2 \alpha_3 \cdots \alpha'_{2j}} = 0 \\ \end{align}
which follow the pattern;

(\gamma^\mu \hat{P}_\mu + mc)_{\alpha_r \alpha'_r}\psi_{\alpha_1 \cdots \alpha'_r \cdots \alpha_{2j}} = 0



(2)

for r = 1, 2, ... 2j. Again, the operator (−γ^{μ}P_{μ} + mc) is a 4 × 4 matrix. The wavefunction Ψ = Ψ(r, t) has components

\psi_{\alpha_1 \alpha_2 \alpha_3 \cdots \alpha_{2j}} (\mathbf{r},t)
and is now a rank2j 4component spinor field, usually symmetric in all bispinor indices, but not necessarily; for example, the spin0 case is antisymmetric. Each index takes the values 1, 2, 3, or 4, so there are 4^{2j} components of the entire spinor field Ψ, although a completely symmetric wavefunction reduces the number of independent components to 2(2j + 1).
Some authors (for example Loide and Saar^{[4]}) use n = 2j, where n is a nonnegative integer (thereby j is a halfinteger or integer), because this helps remove factors of 2.
The above matrix operator contracts with one bispinor index of Ψ at a time (analogous but not equivalent to matrix multiplication), so some properties of the Dirac equation also apply to the BW equations:


E^2 = (pc)^2 + (mc^2)^2
The components for a totally symmetric wavefunction are explicitly:^{[3]}

\Psi = \begin{pmatrix} \psi_{1 \alpha_2 \alpha_3 \cdots \alpha_{2j}} \\ \psi_{2 \alpha_2 \alpha_3 \cdots \alpha_{2j}} \\ \psi_{3 \alpha_2 \alpha_3 \cdots \alpha_{2j}} \\ \psi_{4 \alpha_2 \alpha_3 \cdots \alpha_{2j}} \\ \end{pmatrix}
where the indices are selected so that: α_{2} ≤ α_{3} ≤ ... α_{2j}.
Unlike the Dirac equation, which can incorporate the electromagnetic field via minimal coupling (1C), 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 change P_{μ} → P_{μ} − eA_{μ}.^{[7]}^{[8]} An indirect approach to investigate electromagnetic influences of the particle is to derive the electromagnetic fourcurrents currents and multipole moments for the particle, rather than include the interactions in the wave equations themselves.^{[9]}^{[10]}
Coupled equations
Analogous to (1A) and (1B), the BW equations can be written as a set of coupled equations:

(\hat{E} + mc )^{2j} \psi_{1,2}^ = (\boldsymbol{\sigma}\cdot\hat{\mathbf{p}})^ \psi_{3,4}^


(2A)


(\hat{E} + mc )^{2j} \psi_{3,4}^ = (\boldsymbol{\sigma}\cdot\hat{\mathbf{p}})^ \psi_{1,2}^


(2B)

where the notation [2j] denotes the 2j induced spinor or matrix (defined in the next section). Each of ψ_{1,2} and ψ_{3,4} has 2j + 1 independent components.
These can be recombined:^{[3]}

[(E^2  (mc^2)^2)^{2j}  (\hat{\mathbf{p}}^2)^{2j}]\begin{pmatrix} \psi_{1,2}^ \\ \psi_{3,4}^ \end{pmatrix} = 0


(2C)

which upon expanding by the binomial theorem, then factorizing;

[(E^2  (mc^2)^2)  (\hat{\mathbf{p}}^2)]\left( (E^2  (mc^2)^2)^{2j1} + (E^2  (mc^2)^2)^{2j2}\hat{\mathbf{p}}^2 + \cdots (\hat{\mathbf{p}}^2)^{2j1} \right)\begin{pmatrix} \psi_{1,2}^ \\ \psi_{3,4}^ \end{pmatrix} = 0


(2C)

shows that each component of the BW wavefunction also satisfies the Klein–Gordon equation, uniquely. Conversely, the solutions to the Klein–Gordon equation satisfy the BW equations but are not unique.
Modified gamma matrices
If we define the following Kronecker product (denoted ⊗) of 4 × 4 identity matrices (denoted I_{4}), with the γ^{μ} matrix in the rth place of the product,^{[4]}

\gamma_r^\mu = \underbrace{I_4 \otimes I_4 \otimes \cdots}_{r1\,\text{matrices}} \gamma^\mu \cdots \otimes I_4
for r = 1, 2 ... 2j, these equations (2) can also be written:

(\gamma_r^\mu \hat{P}_\mu  mc )\Psi =0


(3)

The γ_{r}^{μ} matrices have dimension 4^{2j} × 4^{2j}. The equations are linear, so adding (3) with respect to the r values gives:

\left(\frac{1}{2j}\sum_{r=1}^{2j}\gamma_r^\mu \hat{P}_\mu  mc \right)\Psi =0


(3A)

where the factor of 1/2j is inserted because the matrix elements ±1, ±i are added 2j times. Subtracting (3), one r from the next r + 1; the wavefunction satisfies:

(\gamma_r^\mu  \gamma_{r+1}^\mu)\hat{P}_\mu\psi=0


(3B)

for r = 1, 2 ... 2j − 1.
JoosWeinberg equation
Introducing a 2(2j + 1) × 2(2j + 1) matrix;^{[11]}

\gamma^{\mu_1 \mu_2 \cdots \mu_{2j}}
symmetric in any two tensor indices, which generalizes the gamma matrices in the Dirac equation,^{[3]}^{[12]} the BW equation takes the form:^{[13]}^{[14]}

[(i\hbar)^{2j}\gamma^{\mu_1 \mu_2 \cdots \mu_{2j}} \partial_{\mu_1}\partial_{\mu_2}\cdots\partial_{\mu_{2j}} + (mc)^{2j}]\Psi = 0
or

[\gamma^{\mu_1 \mu_2 \cdots \mu_{2j}} P_{\mu_1}P_{\mu_2}\cdots P_{\mu_{2j}} + (mc)^{2j}]\Psi = 0



(4)

This is also known as the JoosWeinberg equation (or JW or JWE), after H. Joos and Steven Weinberg, found in the early 1960s.^{[3]}^{[11]}
Induced matrices
Definition
The induced matrices^{[3]} arise from the spinor transformation:

\begin{pmatrix} a & c \\ b & d \\ \end{pmatrix}\begin{pmatrix} \psi_1 \\ \psi_2 \\ \end{pmatrix}=\begin{pmatrix} \chi_1 \\ \chi_2 \\ \end{pmatrix}


(5)

that is:

\begin{align} a\psi_1 + c\psi_2 &= \chi_1, \\ b\psi_1 + d\psi_2 &= \chi_2. \end{align}
The 2j induced matrix arises by expanding:

(a\psi_1 + c\psi_2)^{j+m}(b\psi_1 + d\psi_2)^{jm} = \chi_1^{j+m}\chi_2^{jm}\,,
for m = −j, −j + 1, ... j − 1, j, simplifying, then writing the set of equations in matrix form.
Properties
Two reasons for introducing the induced matrices is the simple correspondence between induced matrices and powers of eigenvalues, and ease of diagonalization.

Eigenvalues
If A is a 2 × 2 matrix, the 2j induced matrix A^{[2j]} has eigenvalues λ_{1}^{j + m}λ_{2}^{j − m} for the same m values as above.

Diagonalization
If the transformation A → B^{−1}AB holds, then B^{[2j]} will diagonalize A^{[2j]}.
Use in the BW formalism
In the above equations (1A), (1B), (2A), (2B):

(\boldsymbol{\sigma}\cdot\hat{\mathbf{p}})^ = (i \hat{\mathbf{p}})^{2j}e^{i\pi\mathbf{J}^{(j)}\cdot\mathbf{n}}


(6)

where matrix indices on the left side are understood to be m, m′ = −j, −j + 1 ... j. The mm′ element of the (2j + 1) × (2j + 1) matrix contains the energy–momentum operators and are given by:

{(\boldsymbol{\sigma}\cdot\hat{\mathbf{p}})^_{mm'} = (1)^{m'm}\sum_{r=\infty}^\infty\frac{(1)^rp_{}^j(\hat{p}_z)^{jm'r}\hat{p}_z^{j+mr}(p_{+})^{m'm+r}}{r!(jm'r)!(j+mr)!(m'm+r)!}\sqrt{(j+m)!(jm)!(j+m')!(jm')!}}


(7)

where n = p/p is a unit vector and J^{(j)} = (J^{(j)}_{1}, J^{(j)}_{2}, J^{(j)}_{3}) is the vector of the Pauli matrices for spin s.^{[15]}
The matrix (σ • p)^{[2j]} has eigenvalues ±p^{2j}. The degeneracy of the eigenvalues are as follows:


+p^{[2j]}

−p^{[2j]}

Integer spin

(j + 1)fold

jfold

Halfinteger spin

(j + ½)fold

(j + ½)fold

Lorentz group structure
Under a proper orthochronous Lorentz transformation x → Λx in Minkowski space, all oneparticle quantum states ψ^{j}_{σ} of spin j with spin zcomponent σ locally transform under some representation D of the Lorentz group:^{[11]}^{[16]}

\psi(x) \rightarrow D(\Lambda) \psi(\Lambda^{1}x)
where D(Λ) is some finitedimensional representation, i.e. a matrix. Here ψ is thought of as a column vector containing components with the allowed values of σ. The quantum numbers j and σ as well as other labels, continuous or discrete, representing other quantum numbers are suppressed. One value of σ may occur more than once depending on the representation. Representations with several possible values for j are considered below.
The irreducible representations are labeled by a pair of halfintegers or integers (A, B). From these all other representations can be built up using a variety of standard methods, like taking tensor products and direct sums. In particular, spacetime itself constitutes a 4vector representation (1/2, 1/2) so that Λ ∈ D'^{(1/2, 1/2)}. To put this into context; Dirac spinors transform under the (1/2, 0) ⊕ (0, 1/2) representation. In general, the (A, B) representation space has subspaces that under the subgroup of spatial rotations, SO(3), transform irreducibly like objects of spin j, where each allowed value:

j = A + B, A + B  1, ..., A  B,
occurs exactly once.^{[17]} In general, tensor products of irreducible representations are reducible; they decompose as direct sums of irreducible representations.
The representation for the BW equations is the choice:^{[7]}

D^\mathrm{BW} = \bigotimes_{r=1}^{2j} \left[ D_r^{(1/2,0)}\oplus D_r^{(0,1/2)}\right]\,.
where each D_{r} is an irreducible representation. This representation does not have definite spin unless j equals 1/2 or 0. One may perform a Clebsch–Gordan 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 D^{BW} representation satisfies field equations.
For the JW equations the choice is:^{[7]}

D^\mathrm{JW} = D^{(j,0)}\oplus D^{(0,j)}\,.
This representation has definite spin j. It turns out that a spin j particle in this representation satisfy field equations too. These equations are very much like the Dirac equations. It is suitable when the symmetries of charge conjugation, time reversal symmetry, and parity are good.
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 KleinGordon equation.
Lagrangian
The Lagrangian which generates equations (2) through the Euler–Lagrange equation (for fields) is not easily found. Methods have been introduced by Guralnik and Kibble, and Larsen and Repko.^{[18]}
One method proposed by Kamefuchi and Takahashi in 1966 was to expand the wavefunctions in terms of 4 × 4 matrices with a required symmetry (conserved properties of the quantum system), then substitute back into the BW equations to yield field equations with that symmetry. From then a Lagrangian can be found by working backwards from the Euler–Lagrange field equations.
D.S. Kaparulin, S.L. Lyakhovich, and A.A. Sharapov take this fundamental approach by starting from symmetries directly, by means of a Poincaré invariant Lagrange anchor.^{[19]} A Lagrange anchor geometrically defines a mapping between fiber bundles, comprising vector bundles, tangent bundles, and the configuration space for the quantum fields. This is less restrictive than a variational formulation (based on the principle of least action) to obtain the equations for the quantum fields.
Formulation in curved spacetime
Following M. Kenmoku,^{[16]} in local Minkowski space, the gamma matrices satisfy the anticommutation relations:

[\gamma^i,\gamma^j]_{+} = 2\eta^{ij}
where η^{ij} = diag(−1, 1, 1, 1) is the Minkowski metric. For the Latin indices here, i, j = 1, 2, 3. In curved spacetime they are similar:

[\gamma^\mu,\gamma^\nu]_{+} = 2g^{\mu\nu}
where the spatial gamma matrices are contracted with the vierbein b_{i}^{μ} to obtain γ^{μ} = b_{i}^{μ} γ^{i}, and g^{μν} = b^{iμ}b_{i}^{ν} is the metric tensor. For the Greek indices; μ, ν = 0, 1, 2, 3.
A covariant derivative for spinors is given by

\mathcal{D}_\mu=\partial_\mu+\Omega_\mu
with the connection Ω given in terms of the spin connection ω by:

\Omega_\mu =\frac{1}{4}\partial_\mu\omega^{ij} (\gamma_i\gamma_j\gamma_j\gamma_i)
The covariant derivative transforms like ψ:

\mathcal{D}_\mu\psi \rightarrow D(\Lambda) \mathcal{D}_\mu \psi
With this setup, equation (2) becomes:

\begin{align} & (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{align}
See also
References
Notes

^ 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.

^ E. Wigner (1937). "On Unitary Representations Of The Inhomogeneous Lorentz Group". Annals of Mathematics 40 (1): 149.

^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} E.A. Jeffery (1978). "Component Minimization of the Bargman–Wigner wavefunction". Australian Journal of Physics (Melbourne: CSIRO). NB: The convention for the four gradient in this article is ∂_{μ} = (∂/∂t, ∇ ), same as the WorldHeritage article. Jeffery's conventions are different: ∂_{μ} = (−i∂/∂t, ∇ ). Also Jeffery uses collects the x and y components of the momentum operator: p_{±} = p_{1} ± ip_{2} = p_{x} ± ip_{y}. The components p_{±} are not to be confused with ladder operators; the factors of ±1, ±i occur from the gamma matrices.

^ ^{a} ^{b} ^{c} ^{d} R.K Loide, I.Ots, R. Saar (2001). "Generalizations of the Dirac equation in covariant and Hamiltonian form". Journal of Physics A: Mathematical and General (Tallinn, Estonia:

^ C.B. Parker (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). p. 1514.

^ H. ShiZhong, R. TuNan, W. Ning, Z. ZhiPeng (2002). "Wavefunctions for Particles with Arbitrary Spin". Beijing, China: International Academic Publishers.

^ ^{a} ^{b} ^{c} T. Jaroszewicz, P.S Kurzepa (1992). "Geometry of spacetime propagation of spinning particles". Annals of Physics (California, USA).

^ C.R. Hagen (1970). "The Bargmann–Wigner method in Galilean relativity". Communications in Mathematical Physics 18 (2). pp. 97–108.

^ Cédric Lorcé (2009). "Electromagnetic Properties for Arbitrary Spin Particles: Part 1 − Electromagnetic Current and Multipole Decomposition". arXiv:0901.4199.

^ Cédric Lorcé (2009). "Electromagnetic Properties for Arbitrary Spin Particles: Part 2 − Natural Moments and Transverse Charge Densities". arXiv:0901.4200.

^ ^{a} ^{b} ^{c} Weinberg, S. (1964). spin"for Any"Feynman Rules . Phys. Rev. 133 (5B): B1318–B1332.

^ Gábor Zsolt Tóth (2012). "Projection operator approach to the quantization of higher spin fields". arXiv:1209.5673.

^ V.V. Dvoeglazov (2003). "Generalizations of the Dirac Equation and the Modified Bargmann–Wigner Formalism". arXiv:hepth/0208159.

^ D. Shay (1968). "A Lagrangian formulation of the Joos–Weinberg wave equations for spinj particles". Il Nuovo Cimento A 57 (2): 210–218.

^ E. Abers (2004). Quantum Mechanics. Addison Wesley.

^ ^{a} ^{b} K. Masakatsu (2012). "Superradiance Problem of Bosons and Fermions for Rotating Black Holes in Bargmann–Wigner Formulation". arXiv:1208.0644.pdf.

^ Weinberg, S (2002), "5", The Quantum Theory of Fields, vol I,

^ M.A. Rodriguez (1984). "Some results about the relationship between Bargmann–Wigner and Gelfand–Yaglom equations". Reports on Mathematical Physics (Madrid, Spain: Elsevier).

^ D. S. Kaparulin, S. L. Lyakhovich, A. A. Sharapov (2012). "Lagrange Anchor for Bargmann–Wigner equations". arXiv:1210.2134.pdf. Bibcode 2012arXiv1210.2134K.
Further reading
Books

Weinberg, S, The Quantum Theory of Fields, vol II

Weinberg, S, The Quantum Theory of Fields, vol III

R. Penrose (2007).
Selected papers

E. N. Lorenz (1941). "A Generalization of the Dirac Equations". PNAS 27 (6): 317–322.

I. I. Guseinov (2012). "Use of group theory and Clifford algebra in the study of generalized Dirac equation for particles with arbitrary spin". arXiv:0805.1856.

V. V. Dvoeglazov (2011). "The modified BargmannWigner formalism for higher spin fields and relativistic quantum mechanics".

D. N. Williams (1965). "Lectures in Theoretical Physics" 7A. University Press of Colorado. pp. 139–172.

H. ShiZhong, Z. PengFei, R. TuNan, Z. YuCan, Z. ZhiPeng (2004). "Projection Operator and Feynman Propagator for a Free Massive Particle of Arbitrary Spin". Communications in Theoretical Physics 41 (03): 405–418.

V. P. Neznamov (2004). "On the theory of interacting fields in FoldyWouthuysen representation". arXiv:hepth/0411050.

H. Stumpf (2004). "Generalized de Broglie–Bargmann–Wigner Equations, a Modern Formulation of de Broglie’s Fusion Theory". Annales de la Fondation Louis de Broglie 29 (Supplement). p. 785.

D. G. C. McKeon, T. N. Sherry (2004). "The Bargmann–Wigner Equations in Spherical Space". arXiv:hepth/0411090.

R. Clarkson, D. G. C. McKeon (2003). "Quantum Field Theory". pp. 61–69.

H. Stumpf (2002). "Eigenstates of Generalized de Broglie–Bargmann–Wigner Equations for Photons with Partonic Substructure". Z. Naturforsch. 57. pp. 726–736.

B. Schroer (1997). "Wigner Representation Theory of the Poincaré Group, Localization , Statistics and the SMatrix". arXiv:hepth/9608092.

V. V. Dvoeglazov (1993). "Lagrangian Formulation of the Joos–Weinberg's 2(2j+1)–theory and Its Connection with the SkewSymmetric Tensor Description". arXiv:hepth/9305141.

E. Elizalde, J.A. Lobo (1980). "From Galileaninvariant to relativistic wave eqautions". Physical Review D 22 (4). p. 884.

D. V. Ahluwalia (1997). "Book Review: The Quantum Theory of Fields Vol. I and II by S. Weinberg". arXiv:physics/9704002.

J. A. Morgan (2004). "Parity and the SpinStatistics Connection". arXiv:physics/0410037.
External links
Relativistic wave equations:

, Wolfram Demonstrations ProjectDirac matrices in higher dimensions

, P. Cahill, K. Cahill, University of New MexicoLearning about spin1 fields

, R.W. Davies, K.T.R. Davies, P. Zory, D.S. Nydick, American journal of physicsField equations for massless bosons from a Dirac–Weinberg formalism

, Martin MojžišQuantum field theory I

, FarzadQassemi, IPM School and Workshop on Cosmology, IPM, Tehran, IranThe Bargmann–Wigner Equation: Field equation for arbitrary spin
Lorentz groups in relativistic quantum physics:

, indiana.eduRepresentations of Lorentz Group

, mcgill.caAppendix C: Lorentz group and the Dirac algebra

, D. E. Soper, University of Oregon, 2011The Lorentz Group, Relativistic Particles, and Quantum Mechanics

, J. Maciejko, Stanford UniversityRepresentations of Lorentz and Poincaré groups

, K. Drake, M. Feinberg, D. Guild, E. Turetsky, 2009Representations of the Symmetry Group of Spacetime
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