Translational symmetries of quadratic lagrangians
In this paper we show that a quadratic lagrangian, with no constraints, containing ordinary time derivatives up to the order $m$ of $N$ dynamical variables, has $2mN$ symmetries consisting in the translation of the variables with solutions of the equations of motion. We construct explicitly the generators of these transformations and prove that they satisfy the Heisenberg algebra. We also analyse other specific cases which are not included in our previous statement: the Klein-Gordon lagrangian, $N$ Fermi oscillators and the Dirac lagrangian. In the first case, the system is described by an equation involving partial derivatives, the second case is described by Grassmann variables and the third shows both features. Furthermore, the Fermi oscillator and the Dirac field lagrangians lead to second class constraints. We prove that also in these last two cases there are translational symmetries and we construct the algebra of the generators. For the Klein-Gordon case we find a continuum version of the Heisenberg algebra, whereas in the other cases, the Grassmann generators satisfy, after quantization, the algebra of the Fermi creation and annihilation operators.