Symmetries of relativistic world lines

Abstract

Symmetries are essential for a consistent formulation of many quantum systems. In this paper we discuss a previously unnoticed symmetry, which is present for any Lagrangian term that involveṡ x 2 . As a basic model that incorporates the fundamental symmetries of quantum gravity and string theory, we consider the Lagrangian action of the relativistic point particle. A path integral quantization for this seemingly simple system has for long presented notorious problems. Here we show that those problems are overcome by taking into account the newly discovered additional symmetry, leading directly to the exact Klein-Gordon propagator.

The stepwise path integral of the relativistic point particle

“…The generalization of the results (19) and (23) for the n-step propagator is straightforward and it follows from the same arguments already presented in subsection II C, if one considers the following adjustments…”

“…(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.…”

“…(b) Local Lorentz invariance: This means that the Lagrangian is invariant under local rotations in D dimensions of the vector (dx)/(dλ) at any point along the trajectory. A formal argument on why this symmetry, which is not a classical gauge symmetry, is actually important in this given context was given in [12]. (c) Weyl invariance: This means that the Lagrangian does not depend on the way that λ parametrizes a path P. The change to any other functionλ(λ) would leave the Lagrangian invariant.…”

A hidden constraint on the Hamiltonian formulation of relativistic worldlines

“…We have presented a new constraint (12) for the RPP and shown that it reflects all the nice features demanded in table II. In particular, it generates a non-trivial local symmetry of the action without altering the classical equations of motion of the system.…”

“…Thus, δx µ reflects the spirit of the velocity rotations in the Lagrangian formulation [12,13]. The constraint further generates a variation of the action…”