Lagrangian and Hamiltonian formulation of relativistic particle mechanics

Abstract

It is shown that any (relativistic, action-at-a-distance theory formulated in terms of a (relativistically invariant) Fokker action, which is invariant under any change of the world-line parameter, also can be given a Lagrangian formulation. The parametrization invariance makes it possible to introduce a common parameter which also can be identified with time. The resulting Lagrangians are nonlocal in time and therefore the applicability of the action principle requires a generalized variation method which is developed here. The resulting generators are expected to generate the corresponding variations in a Poisson-bracket sense, i.e., Hamiltonian formulation of the theory is expected to exist. However, in order to check this it is necessary to (formally) solve the equations of motion, as one needs Poisson brackets for unequal time due to the nonlocality of the generators. But since the equations of motion themselves are nonlocal in time, the Newtonian initial data will not yield a unique solution. Thus, in order to retain the Newtonian degrees of freedom one needs a selection principle as a subsidiary condition. Such a principle has been proposed which states that one has to choose the solution which has a nonrelativistic limit. In the case of a general vector interaction the class of solutions which is obtained by an iterative method (which starts from straight lines) is considered. These solutions are uniquely determined by the Newtonian initial data, but they are applicable only to scattering processes. By choosing the asymptotic straight lines as canonical variables it is explicitly shown that the Poincar\'e generators fulfill the Lie algebra of the Poincar\'e group and that the physical positions transform as Lorentz vectors. It is also shown that the physical positions cannot be chosen as canonical variables. Quantization is discussed and a general argument is put forth which states that the above solutions cannot be quantized by imposing the canonical commutation relations on the canonical variables.