Lorentz - covariant spectrum of single - particle states and their field theory

In order to understand the number of degrees of freedom we need to include in a Lorentz-invariant theory, we need to develop the representation theory of the Poincaré symmetry. The Poincaré symmetry consists of two sets of symmetries of the Minkowski spacetime. Space-time translations generated by the energy-momentum four vector P , and the Lorentz transformations (rotation and boost) generated by the operators M . I can’t remember the commutation relations, and I always work them out using the analogy to the single-particle quantum mechanics. The translation generates are energy and momentum Pμ = i∂μ, and the boost and rotation M = xp − xp = i(x∂ − xν∂μ).∗ The commutators are [P , P ν ] = 0, (1)