Lorentz covariant tensor description of Weinberg–Joos states
The six-component spin-1 representation space,
can be labeled by a pair of anti-symmetric Lorentz indexes, [αβ], meaning that it transforms as an antisymmetric Lorentz tensor of second rank ![{\displaystyle B_{[\alpha \beta ]},}](//wikimedia.org/api/rest_v1/media/math/render/svg/e2df67921a8f9d27184d89af3f5e97ef3a307475)
i.e.
![{\displaystyle B_{[\alpha \beta ]}\sim D^{(1,0)}\oplus D^{(0,1)}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/cc5456d65d6871ca949b1b65f0ccb20ef19db51c)
The j-fold Kronecker product T[α1β1]...[αjβj] of B[αβ]
![{\displaystyle T_{[\alpha _{1}\beta _{1}]\cdots [\alpha _{j}\beta _{j}]}=B_{[\alpha _{1}\beta _{1}]}\otimes \cdots \otimes B_{[\alpha _{j}\beta _{j}]}=\bigotimes _{i=1}^{j}B_{[\alpha _{i}\beta _{i}]},}](//wikimedia.org/api/rest_v1/media/math/render/svg/36d9cc88a1e026a1672770a56d22f1124ee91f01) | | 8A |
decomposes into a finite series of Lorentz-irreducible representation spaces according to
and necessarily contains a 
sector. This sector can instantly be identified by means of a momentum independent projector operator P(j,0), designed on the basis of C(1), one of the Casimir elements (invariants)[7] of the Lie algebra of the Lorentz group, which are defined as,
![{\displaystyle {\begin{cases}\left[C^{(1)}\right]_{AB}={\frac {1}{4}}{\left[M^{\mu \nu }\right]_{A}}^{C}\left[M_{\mu \nu }\right]_{CB}\\[6pt]\left[C^{(2)}\right]_{AB}={\frac {1}{4}}\varepsilon _{\mu \nu \lambda \eta }{\left[M^{\mu \nu }\right]_{A}}^{C}\left[M^{\lambda \eta }\right]_{CB}\end{cases}}\qquad A,B,C=1,\ldots ,(2j_{1}+1)(2j_{2}+1)}](//wikimedia.org/api/rest_v1/media/math/render/svg/ee168de6af8d6deb14773c12eaae3c150856774d) | | 8B |
where Mμν are constant (2j1+1)(2j2+1) × (2j1+1)(2j2+1) matrices defining the elements of the Lorentz algebra within the 
representations. The Capital Latin letter labels indicate[8] the finite dimensionality of the representation spaces under consideration which describe the internal angular momentum (spin) degrees of freedom.
The representation spaces are eigenvectors to C(1) in (8B) according to,
![{\displaystyle C^{(1)}\left[D^{(j_{1},j_{2})}\oplus D^{(j_{2},j_{1})}\right]=\left(j_{1}(j_{1}+1)+j_{2}(j_{2}+1)\right)\left[D^{(j_{1},j_{2})}\oplus D^{(j_{2},j_{1})}\right],}](//wikimedia.org/api/rest_v1/media/math/render/svg/ef7a943bdebce802e6d92548ca291ad424fe7902)
Here we define:
to be the C(1) eigenvalue of the sector. Using this notation we define the projector operator, P(j,0) in terms of C(1):[8]
![{\displaystyle {\left[P^{(j,0)}\right]_{\left[\alpha _{1}\beta _{1}\right]\cdots \left[\alpha _{j}\beta _{j}\right]}}^{\left[\rho _{1}\sigma _{1}\right]\cdots \left[\rho _{j}\sigma _{j}\right]}={\left[\prod _{k,l}\left({\frac {C^{(1)}-\lambda _{(j_{k},j_{l})}^{(1)}}{\lambda _{(j,\,0)}^{(1)}-\lambda _{(j_{k},j_{l})}^{(1)}}}\right)\right]_{\left[\alpha _{1}\beta _{1}\right]\cdots \left[\alpha _{j}\beta _{j}\right]}}^{\left[\rho _{1}\sigma _{1}\right]\cdots \left[\rho _{j}\sigma _{j}\right]}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/76cb3e46f48fcb0b4684841cd9e7ff8af4493311) | | 8C |
Such projectors can be employed to search through T[α1β1]...[αjβj] for 
and exclude all the rest. Relativistic second order wave equations for any j are then straightforwardly obtained in first identifying the sector in T[α1β1]...[αjβj] in (8A) by means of the Lorentz projector in (8C) and then imposing on the result the mass shell condition.
This algorithm is free from auxiliary conditions. The scheme also extends to half-integer spins, 
in which case the Kronecker product of T[α1β1]...[αjβj] with the Dirac spinor,
has to be considered. The choice of the totally antisymmetric Lorentz tensor of second rank, B[αiβi], in the above equation (8A) is only optional. It is possible to start with multiple Kronecker products of totally symmetric second rank Lorentz tensors, Aαiβi. The latter option should be of interest in theories where high-spin Joos–Weinberg fields preferably couple to symmetric tensors, such as the metric tensor in gravity.
An Example
Source:[8]
The
transforming in the Lorenz tensor spinor of second rank,
![{\displaystyle \psi _{[\mu \nu ]}=[(1,0)\oplus (0,1)]\otimes \left[\left({\tfrac {1}{2}},0\right)\oplus \left(0,{\tfrac {1}{2}}\right)\right].}](//wikimedia.org/api/rest_v1/media/math/render/svg/8d31cfa95aa6f816154be145d07e1fc6b10e9e9e)
The Lorentz group generators within this representation space are denoted by ![{\displaystyle \left[M_{\mu \nu }^{ATS}\right]_{[\alpha \beta ][\gamma \delta ]},}](//wikimedia.org/api/rest_v1/media/math/render/svg/924d08bbd7727442a97db4c462acbec6a8d2c513)
and given by:
![{\displaystyle \left[M_{\mu \nu }^{ATS}\right]_{[\alpha \beta ][\gamma \delta ]}=\left[M_{\mu \nu }^{AT}\right]_{[\alpha \beta ][\gamma \delta ]}{\mathbf {1} }^{S}+{\mathbf {1} }_{[\alpha \beta ][\gamma \delta ]}\,\,\left[M_{\mu \nu }^{S}\right],}](//wikimedia.org/api/rest_v1/media/math/render/svg/f3922a10e4113fe5dfc8849bd99232f128793df6)
![{\displaystyle \mathbf {1} _{[\alpha \beta ][\gamma \delta ]}={\tfrac {1}{2}}\left(g_{\alpha \gamma }g_{\beta \delta }-g_{\alpha \delta }g_{\beta \gamma }\right),}](//wikimedia.org/api/rest_v1/media/math/render/svg/24be8781bc2e892d4c0df53cd422e4d91553bc03)
![{\displaystyle M_{\mu \nu }^{S}={\tfrac {1}{2}}\sigma _{\mu \nu }={\frac {i}{4}}[\gamma _{\mu },\gamma _{\nu }],}](//wikimedia.org/api/rest_v1/media/math/render/svg/fd7320e6ac213c3aff5eaf67224904d8017fb756)
where 1[αβ][γδ] stands for the identity in this space, 1S and MSμν are the respective unit operator and the Lorentz algebra elements within the Dirac space, while γμ are the standard gamma matrices. The [MATμν][αβ][γδ] generators express in terms of the generators in the four-vector,
![{\displaystyle \left[M_{\mu \nu }^{V}\right]_{\alpha \beta }=i\left(g_{\alpha \mu }g_{\beta \nu }-g_{\alpha \nu }g_{\beta \mu }\right),}](//wikimedia.org/api/rest_v1/media/math/render/svg/bdf665384004c4755fb1cf32566be4ff219f14d2)
as
![{\displaystyle \left[M_{\mu \nu }^{AT}\right]_{[\alpha \beta ][\gamma \delta ]}=-2\cdot {\mathbf {1} _{[\alpha \beta ]}}^{[\kappa \sigma ]}{\left[M_{\mu \nu }^{V}\right]_{\sigma }}^{\rho }{\mathbf {1} }_{[\rho \kappa ][\gamma \delta ]}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/45bfea7cf82914276c42ebec8492cf60b2f2582d)
Then, the explicit expression for the Casimir invariant C(1) in (8B) takes the form,
![{\displaystyle \left[C^{(1)}\right]_{[\alpha \beta ][\gamma \delta ]}=-{\frac {1}{8}}\left(\sigma _{\alpha \beta }\sigma _{\gamma \delta }-\sigma _{\gamma \delta }\sigma _{\alpha \beta }-22\cdot \mathbf {1} _{[\alpha \beta ][\gamma \delta ]}\right),}](//wikimedia.org/api/rest_v1/media/math/render/svg/66f7523051716976adec818e3c22d4258e3f1acb)
and the Lorentz projector on (3/2,0)⊕(0,3/2) is given by,
![{\displaystyle \left[P^{\left({\frac {3}{2}},0\right)}\right]_{[\alpha \beta ][\gamma \delta ]}={\frac {1}{8}}\left(\sigma _{\alpha \beta }\sigma _{\gamma \delta }+\sigma _{\gamma \delta }\sigma _{\alpha \beta }\right)-{\frac {1}{12}}\sigma _{\alpha \beta }\sigma _{\gamma \delta }.}](//wikimedia.org/api/rest_v1/media/math/render/svg/b1b90b1d999bf86e1634cf5097cabb1251b8b956)
In effect, the (3/2,0)⊕(0,3/2) degrees of freedom, denoted by
![{\displaystyle \left[w_{\pm }^{\left({\frac {3}{2}},0\right)}\left({\mathbf {p} },{\tfrac {3}{2}},\lambda \right)\right]^{[\gamma \delta ]}}](//wikimedia.org/api/rest_v1/media/math/render/svg/3f4423a58eadeaa65fb552cf6f16756b335ef2b4)
are found to solve the following second order equation,
![{\displaystyle \left({\left[P^{\left({\frac {3}{2}},0\right)}\right]^{[\alpha \beta ]}}_{[\gamma \delta ]}p^{2}-m^{2}\cdot {{\mathbf {1} }^{[\alpha \beta ]}}_{[\gamma \delta ]}\right)\left[w_{\pm }^{\left({\frac {3}{2}},0\right)}\left({\mathbf {p} },{\tfrac {3}{2}},\lambda \right)\right]^{[\gamma \delta ]}=0.}](//wikimedia.org/api/rest_v1/media/math/render/svg/883346b13b06146eb6c6d2dec745447df7e4d3e3)
Expressions for the solutions can be found in.[8]
![{\displaystyle [(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}](http://wikimedia.org/api/rest_v1/media/math/render/svg/c77b541c6249118153d93dc426d8bbd313cae1ad)
![{\displaystyle [\gamma ^{\mu _{1}\mu _{2}\cdots \mu _{2j}}P_{\mu _{1}}P_{\mu _{2}}\cdots P_{\mu _{2j}}+(mc)^{2j}]\Psi =0}](http://wikimedia.org/api/rest_v1/media/math/render/svg/3ec0dbcee1521316f26936a442c471a531d00542)
![{\displaystyle B_{[\alpha \beta ]},}](http://wikimedia.org/api/rest_v1/media/math/render/svg/e2df67921a8f9d27184d89af3f5e97ef3a307475)
![{\displaystyle B_{[\alpha \beta ]}\sim D^{(1,0)}\oplus D^{(0,1)}.}](http://wikimedia.org/api/rest_v1/media/math/render/svg/cc5456d65d6871ca949b1b65f0ccb20ef19db51c)
![{\displaystyle T_{[\alpha _{1}\beta _{1}]\cdots [\alpha _{j}\beta _{j}]}=B_{[\alpha _{1}\beta _{1}]}\otimes \cdots \otimes B_{[\alpha _{j}\beta _{j}]}=\bigotimes _{i=1}^{j}B_{[\alpha _{i}\beta _{i}]},}](http://wikimedia.org/api/rest_v1/media/math/render/svg/36d9cc88a1e026a1672770a56d22f1124ee91f01)
![{\displaystyle {\begin{cases}\left[C^{(1)}\right]_{AB}={\frac {1}{4}}{\left[M^{\mu \nu }\right]_{A}}^{C}\left[M_{\mu \nu }\right]_{CB}\\[6pt]\left[C^{(2)}\right]_{AB}={\frac {1}{4}}\varepsilon _{\mu \nu \lambda \eta }{\left[M^{\mu \nu }\right]_{A}}^{C}\left[M^{\lambda \eta }\right]_{CB}\end{cases}}\qquad A,B,C=1,\ldots ,(2j_{1}+1)(2j_{2}+1)}](http://wikimedia.org/api/rest_v1/media/math/render/svg/ee168de6af8d6deb14773c12eaae3c150856774d)
![{\displaystyle C^{(1)}\left[D^{(j_{1},j_{2})}\oplus D^{(j_{2},j_{1})}\right]=\left(j_{1}(j_{1}+1)+j_{2}(j_{2}+1)\right)\left[D^{(j_{1},j_{2})}\oplus D^{(j_{2},j_{1})}\right],}](http://wikimedia.org/api/rest_v1/media/math/render/svg/ef7a943bdebce802e6d92548ca291ad424fe7902)
![{\displaystyle {\left[P^{(j,0)}\right]_{\left[\alpha _{1}\beta _{1}\right]\cdots \left[\alpha _{j}\beta _{j}\right]}}^{\left[\rho _{1}\sigma _{1}\right]\cdots \left[\rho _{j}\sigma _{j}\right]}={\left[\prod _{k,l}\left({\frac {C^{(1)}-\lambda _{(j_{k},j_{l})}^{(1)}}{\lambda _{(j,\,0)}^{(1)}-\lambda _{(j_{k},j_{l})}^{(1)}}}\right)\right]_{\left[\alpha _{1}\beta _{1}\right]\cdots \left[\alpha _{j}\beta _{j}\right]}}^{\left[\rho _{1}\sigma _{1}\right]\cdots \left[\rho _{j}\sigma _{j}\right]}.}](http://wikimedia.org/api/rest_v1/media/math/render/svg/76cb3e46f48fcb0b4684841cd9e7ff8af4493311)
![{\displaystyle \psi _{[\mu \nu ]}=[(1,0)\oplus (0,1)]\otimes \left[\left({\tfrac {1}{2}},0\right)\oplus \left(0,{\tfrac {1}{2}}\right)\right].}](http://wikimedia.org/api/rest_v1/media/math/render/svg/8d31cfa95aa6f816154be145d07e1fc6b10e9e9e)
![{\displaystyle \left[M_{\mu \nu }^{ATS}\right]_{[\alpha \beta ][\gamma \delta ]},}](http://wikimedia.org/api/rest_v1/media/math/render/svg/924d08bbd7727442a97db4c462acbec6a8d2c513)
![{\displaystyle \left[M_{\mu \nu }^{ATS}\right]_{[\alpha \beta ][\gamma \delta ]}=\left[M_{\mu \nu }^{AT}\right]_{[\alpha \beta ][\gamma \delta ]}{\mathbf {1} }^{S}+{\mathbf {1} }_{[\alpha \beta ][\gamma \delta ]}\,\,\left[M_{\mu \nu }^{S}\right],}](http://wikimedia.org/api/rest_v1/media/math/render/svg/f3922a10e4113fe5dfc8849bd99232f128793df6)
![{\displaystyle \mathbf {1} _{[\alpha \beta ][\gamma \delta ]}={\tfrac {1}{2}}\left(g_{\alpha \gamma }g_{\beta \delta }-g_{\alpha \delta }g_{\beta \gamma }\right),}](http://wikimedia.org/api/rest_v1/media/math/render/svg/24be8781bc2e892d4c0df53cd422e4d91553bc03)
![{\displaystyle M_{\mu \nu }^{S}={\tfrac {1}{2}}\sigma _{\mu \nu }={\frac {i}{4}}[\gamma _{\mu },\gamma _{\nu }],}](http://wikimedia.org/api/rest_v1/media/math/render/svg/fd7320e6ac213c3aff5eaf67224904d8017fb756)
![{\displaystyle \left[M_{\mu \nu }^{V}\right]_{\alpha \beta }=i\left(g_{\alpha \mu }g_{\beta \nu }-g_{\alpha \nu }g_{\beta \mu }\right),}](http://wikimedia.org/api/rest_v1/media/math/render/svg/bdf665384004c4755fb1cf32566be4ff219f14d2)
![{\displaystyle \left[M_{\mu \nu }^{AT}\right]_{[\alpha \beta ][\gamma \delta ]}=-2\cdot {\mathbf {1} _{[\alpha \beta ]}}^{[\kappa \sigma ]}{\left[M_{\mu \nu }^{V}\right]_{\sigma }}^{\rho }{\mathbf {1} }_{[\rho \kappa ][\gamma \delta ]}.}](http://wikimedia.org/api/rest_v1/media/math/render/svg/45bfea7cf82914276c42ebec8492cf60b2f2582d)
![{\displaystyle \left[C^{(1)}\right]_{[\alpha \beta ][\gamma \delta ]}=-{\frac {1}{8}}\left(\sigma _{\alpha \beta }\sigma _{\gamma \delta }-\sigma _{\gamma \delta }\sigma _{\alpha \beta }-22\cdot \mathbf {1} _{[\alpha \beta ][\gamma \delta ]}\right),}](http://wikimedia.org/api/rest_v1/media/math/render/svg/66f7523051716976adec818e3c22d4258e3f1acb)
![{\displaystyle \left[P^{\left({\frac {3}{2}},0\right)}\right]_{[\alpha \beta ][\gamma \delta ]}={\frac {1}{8}}\left(\sigma _{\alpha \beta }\sigma _{\gamma \delta }+\sigma _{\gamma \delta }\sigma _{\alpha \beta }\right)-{\frac {1}{12}}\sigma _{\alpha \beta }\sigma _{\gamma \delta }.}](http://wikimedia.org/api/rest_v1/media/math/render/svg/b1b90b1d999bf86e1634cf5097cabb1251b8b956)
![{\displaystyle \left[w_{\pm }^{\left({\frac {3}{2}},0\right)}\left({\mathbf {p} },{\tfrac {3}{2}},\lambda \right)\right]^{[\gamma \delta ]}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/3f4423a58eadeaa65fb552cf6f16756b335ef2b4)
![{\displaystyle \left({\left[P^{\left({\frac {3}{2}},0\right)}\right]^{[\alpha \beta ]}}_{[\gamma \delta ]}p^{2}-m^{2}\cdot {{\mathbf {1} }^{[\alpha \beta ]}}_{[\gamma \delta ]}\right)\left[w_{\pm }^{\left({\frac {3}{2}},0\right)}\left({\mathbf {p} },{\tfrac {3}{2}},\lambda \right)\right]^{[\gamma \delta ]}=0.}](http://wikimedia.org/api/rest_v1/media/math/render/svg/883346b13b06146eb6c6d2dec745447df7e4d3e3)