Lorentz Group Projector Technique for Decomposing Reducible Representations and Applications to High Spins

The momentum-independent Casimir operators of the homogeneous spin-Lorentz group are employed in the construction of covariant projector operators, which can decompose anyone of its reducible finite-dimensional representation spaces into irreducible components. One of the benefits from such operators is that any one of the finite-dimensional carrier spaces of the Lorentz group representations can be equipped with Lorentz vector indices because any such space can be embedded in a Lorentz tensor of a properly-designed rank and then be unambiguously found by a projector. In particular, all the carrier spaces of the single-spin-valued Lorentz group representations, which so far have been described as 2 ( 2 j + 1 ) column vectors, can now be described in terms of Lorentz tensors for bosons or Lorentz tensors with the Dirac spinor component, for fermions. This approach facilitates the construct of covariant interactions of high spins with external fields in so far as they can be obtained by simple contractions of the relevant S O ( 1 , 3 ) indices. Examples of Lorentz group projector operators for spins varying from 1 / 2 –2 and belonging to distinct product spaces are explicitly worked out. The decomposition of multiple-spin-valued product spaces into irreducible sectors suggests that not only the highest spin, but all the spins contained in an irreducible carrier space could correspond to physical degrees of freedom.

CLASSIFYING REPORTED AND "MISSING" RESONANCES ACCORDING TO THEIR P and C PROPERTIES

The Hilbert space ℋ3q of the three quarks with one excited quark is decomposed into Lorentz group representations. It is shown that the quantum numbers of the reported and "missing" resonances fall apart and populate distinct representations that differ by their parity or/and charge conjugation properties. In this way, reported and "missing" resonances become distinguishable. For example, resonances from the full listing reported by the Particle Data Group are accommodated by Rarita–Schwinger (RS) type representations [Formula: see text] with k=1, 3, and 5, the highest spin states being J=3/2-, 7/2+, and 11/2+, respectively. In contrast to this, most of the "missing" resonances fall into the opposite parity RS fields of highest-spins 5/2-, 5/2+, and 9/2+, respectively. Rarita–Schwinger fields with physical resonances as lower-spin components can be treated as a whole without imposing auxiliary conditions on them. Such fields do not suffer the Velo–Zwanziger problem but propagate causally in the presence of electromagnetic fields. The pathologies associated with RS fields arise basically because of the attempt to use them to describe isolated spin-J=k+½ states, rather than multispin-parity clusters. The positions of the observed RS clusters and their spacing are well explained trough the interplay between the rotational-like [Formula: see text]-rule and a Balmer-like [Formula: see text]-behavior.