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

• Main Authors: Victor Miguel Banda Guzmán, Mariana Kirchbach
• Format: Article
• Language: English
• Published: MDPI AG 2019-08-01
• Series: Universe
• Subjects: homogenous Lorentz group; high spins; covariant projectors; decomposition of tensor products
• Online Access: https://www.mdpi.com/2218-1997/5/8/184

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 <inline-formula> <math display="inline"> <semantics> <mrow> <mn>2</mn> <mo>(</mo> <mn>2</mn> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> 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 <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>O</mi> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> indices. Examples of Lorentz group projector operators for spins varying from <inline-formula> <math display="inline"> <semantics> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>&#8722;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.