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 Klein–Gordon equation.
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 i.e.
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,
(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,
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]
(8C)
Such projectors can be employed to search through Tα1β1]...[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 Tα1β1]...[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 Tα1β1]...[αjβj with the Dirac spinor,
has to be considered. The choice of the totally antisymmetric Lorentz tensor of second rank, Bαiβi, 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.
transforming in the Lorenz tensor spinor of second rank,
The Lorentz group generators within this representation space are denoted by and given by:
where 1αβ][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 MATμναβ][γδ generators express in terms of the generators in the four-vector,
as
Then, the explicit expression for the Casimir invariant C(1) in (8B) takes the form,
and the Lorentz projector on (3/2,0)⊕(0,3/2) is given by,
In effect, the (3/2,0)⊕(0,3/2) degrees of freedom, denoted by
are found to solve the following second order equation,
^
abE.A. Jeffery (1978).
"Component Minimization of the Bargman–Wigner wavefunction". Australian Journal of Physics. 31 (2). Melbourne: CSIRO: 137.
Bibcode:
1978AuJPh..31..137J.
doi:10.1071/ph780137. NB: The convention for the
four-gradient in this article is ∂μ = (∂/∂t, ∇
), same as the Wikipedia article. Jeffery's conventions are different: ∂μ = (−i∂/∂t, ∇
). Also Jeffery uses collects the x and y components of the momentum operator: p± = p1 ± ip2 = px ± ipy. The components p± are not to be confused with
ladder operators; the factors of ±1, ±i occur from the
gamma matrices.
^
abcdE. G. Delgado Acosta; V. M. Banda Guzmán; M. Kirchbach (2015). "Bosonic and fermionic Weinberg-Joos (j,0) ⊕ (0,j) states of arbitrary spins as Lorentz tensors or tensor-spinors and second-order theory". The European Physical Journal A. 51 (3): 35.
arXiv:1503.07230.
Bibcode:
2015EPJA...51...35D.
doi:
10.1140/epja/i2015-15035-x.
S2CID118590440.