A disadvantage is the necessity to impose on such carrier space conditions, termed as “auxiliary”, and supposed to remove the unwanted spin content, an expectation that is met for free fields, but becomes problematic at the level of interactions. The tensor basis characteristic for the former case represents a significant advantage over the latter as it enables construction of covariant interactions with external fields by simple contractions of the S O ( 1, 3 ) indices. In the literature, there are two qualitatively distinct kinds of finite-dimensional representations of the Lorentz group in use, those containing multiple spins as parts of reducible tensor products and those containing one sole spin and represented by irreducible column vectors. The key point in all the methods employed for high-spin description concerns the choice for the representation of the Lorentz group that embeds the spin of interest. Each one of them has some advantages over the others, which however are as a rule gained at the cost of some specific problems. They are associated with the names of Fierz and Pauli (FP), Rarita and Schwinger (RS), Laporte and Uhlenbeck (LU), Cap and Donnert (CD), Bargmann and Wigner (BW), as well as to those of Joos and Weinberg (JW). The traditional methods in the description of high-spin fields were developed in the period between 19 (see for recent reviews and for a standard textbook) and are based on the use of carrier (representation) spaces of finite-dimensional representations of the homogeneous Lorentz group. In addition, high spins are fundamental to the physics of rotating black holes, not to forget gravitational interactions between high-spin fermions. At hadron colliders, they can emerge as intermediate resonances in a variety of processes, while in gravity, deformations of the metric tensor caused by its coupling to high-spin bosons are of interest.
In the physics of hadrons, such fields appear as real resonances whose spins can vary from 1 / 2– 17 / 2 for baryons and from 0–6 for mesons. Particles of high-spins j ≥ 1, be they massive or mass-less, play a significant role in field theories.
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. Examples of Lorentz group projector operators for spins varying from 1 / 2–2 and belonging to distinct product spaces are explicitly worked out. 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. 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. 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. 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.
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