Path integral of the relativistic point particle in Minkowski space

Abstract

In this article, we analyze the fundamental global and local symmetries involved in the action for the free relativistic point particle. Moreover, we identify a hidden local symmetry, whose explicit consideration and factorization utilizing of a Fujikawa prescription, leads to the construction of relativistic propagators that satisfy the Chapman-Kolmogorov identity. By means of a detailed topological analysis, we find three different relativistic propagators (orthochronous, space-like, and Feynman) which are obtained from the exclusive integration of paths within different sectors in Minkowski space. Finally, the connection of this approach to the Feynman checkerboard construction is explored. Contents I. Introduction A. The context B. The relativistic point particle C. General considerations on the explicit form of the PI measure II. The propagator A. Action B. The two step propagator C. The N + 1 step propagator D. Space-like, time-like, orthochronous, and the Feynman propagator E. Comparison to the propagators of the Klein-Gordon field F. Without spatial flip symmetry and Feynmans' checkerboard G. Higher dimensional generalization of the propagator H. Higher dimensional generalization of the checkerboard? III. Conclusion Acknowlegements Appendix References

Non-perturbative Quantum Propagators in Bounded Spaces

“…It is important to highlight some prior studies based on a similar philosophy. The Path Decomposition Expansion of [26][27][28][29] splits up the path integral representation of the Green function associated to the kernel in terms of contributions from trajectories in different spatial regions; in [30,31] the Multi-Step Propagator is introduced with the aim of eliminating a possibly dangerous overcounting of certain types of trajectory in the path integral; the effective action of a (relativistic) scalar field in a localised potential is found in [32] by resumming a perturbative expansion of the interaction (in fact our n-hit function is related to an infinite set of intermediate functions that enter this resummation); and finally, Polyakov long ago formulated a representation of the (one-particle-reducible) contributions to the N -point configuration-space amplitudes in φ 3 theory in terms of a relativistic point-particle path integral constrained to pass through the prescribed external points [33].…”

“…For the case of even dimensions, the recurrence relations and dimensional shift formulae imply that it suffices to obtain the n = 1-hit function in D = 2, say. To achieve this we invoke the integral representation (30) which for n = 1 and D = 2 becomes…”