Review of algebras for particle physics

We review two- and four-dimensional spinor representations of the Lorentz group. The two-dimensional spinor representation (SL(2,C)) is well suited for two-component Weyl spinors, but it is not suitable for four-component objects in the vector form. One usually introduces two-index objects in order to represent electromagnetism either as a four potential or as a Riemann-Silberstein vector. On the other hand, four-dimensional representation of the Lorentz group admits vector and matrix forms of all three objects: four-vectors, Riemann-Silberstein vectors and four-component Weyl spinors. This representation has a flexibility of regarding electromagnetism either as a four-component four-vector (four potential) or as a four-component Riemann-Silberstein vector. Although, two- and four-dimensional representations are equivalent in many respects, one can write various compound and hybrid objects transforming as four-vectors, scalars and four-dimensional scalars that cannot be written in two-dimensions. It becomes more clear in the four-dimensional representation that, for the sake of the completeness and symmetry of the formalism, we have to introduce two more types of transformations, and accordingly two more objects. Under the Lorentz boosts, one of them transforms as a four-vector and the other one as a Riemann-Silberstein vector. But, under the rotations both of them manifest a rather odd behavior. We call these exotic objects W and H and show that their duals are defined by means of the Minkowski metric, hence their scalar products are Lorentz invariant quantities.