A hidden constraint on the Hamiltonian formulation of relativistic worldlines

Abstract

Gauge theories with general covariance are particularly reluctant to quantization. We discuss the example of the Hamiltonian formulation of the relativistic point particle that, despite its apparent simplicity, is of crucial importance since a number of point particle systems can be cast into this form on a higher dimensional Rindler background, as recently pointed out by Hojman. It is shown that this system can be equipped with a hidden local, symmetry generating, constraint which on the one hand does not bother the classical evolution and on the other hand simplifies the realization of the path integral quantization. Even though the positive impact of the hidden symmetry is more evident in the Lagrangian version of the theory, it is still present through the suggested Hamiltonian constraint.

The stepwise path integral of the relativistic point particle

“…(b) Local Lorentz invariance: This means that the Lagrangian is invariant under local rotations and boosts of the vector (dx µ )/(dλ) at any point along the trajectory. A formal argument on why this symmetry, which is not a classical gauge symmetry, is important in this given context was presented in [21][22][23]. In [21] it was further shown in the Hamiltonian action formulation that there is a non-trivial constraint associated with this symmetry.…”

“…A formal argument on why this symmetry, which is not a classical gauge symmetry, is important in this given context was presented in [21][22][23]. In [21] it was further shown in the Hamiltonian action formulation that there is a non-trivial constraint associated with this symmetry.…”

“…A naive, straightforward definition of a path integral for the relativistic point particle fails to satisfy the Kolmogorov condition for transition probability amplitudes. As shown in [21][22][23], this is due to the overcounting arising from the symmetries b), and c), that needs to be properly factored out either by geometric considerations [21,22] or by a group theoretical analysis involving the Fadeev-Popov method [23].…”

“…• A PI over the Hamiltonian action and the corresponding constraint analysis [21] • A direct PI in Euclidean space [22] • A formal functional path integral [23] These results were either obtained in Euclidean space or with abstract formal methods, but a detailed analysis of the much richer structure of Minkowski space is missing.…”