Symmetries in physics

A.1 Symmetries in physics

A.1.1 Derivation of the Poincaré algebra

Perhaps the simplest way to derive these is via the infinite-dimensional representation of the generators as differential operators acting on functions of spacetime $ψ (x^{μ})$ :

\begin{matrix} \begin{aligned} P_{μ} & = - i \partial_{μ}, \\ M_{μν} & = i (x_{μ} \partial_{ν} - x_{ν} \partial_{μ}) . \end{aligned} \end{matrix}

(A.1.1)

These should be familiar as the momentum and angular momentum operators from classical and quantum mechanics, generalized to include boosts and the time dimension. Thus, for example,

\begin{matrix} \begin{aligned} [M_{0 i}, P_{j}] ψ (x^{μ}) & = (- i^{2}) [x_{0} \partial_{i} - x_{i} \partial_{0}, \partial_{j}] ψ \\ = [(x_{0} \partial_{i} \partial_{j} - x_{i} \partial_{0} \partial_{j}) - (\partial_{j} (x_{0} \partial_{i}) - \partial_{j} (x_{i} \partial_{0}))] ψ \\ = [(x_{0} \partial_{i} \partial_{j} - x_{i} \partial_{0} \partial_{j}) - (x_{0} \partial_{i} \partial_{j} - η_{ij} \partial_{0} - x_{i} \partial_{0} \partial_{j})] ψ \\ = η_{ij} \partial_{0} ψ \\ = i η_{ij} P_{0} ψ . \end{aligned} \end{matrix}

(A.1.2)

The rest of the commutation relations in Eqs. 2.3.10 and 2.3.12 can be derived similarly.