A naturally regularized path integral formalism

Abstract

A minimal uncertainty in position measurement comes into play if, for instance, one incorporates gravitational interaction of photon and electron in the Heisenberg's Electron Microscope Gedanken Experiment. This interaction modifies the Heisenberg's standard uncertainty principle (HUP) to the so-called Generalized (Gravitational) Uncertainty Principle (GUP). The finite resolution of spacetime structure (through a minimal uncertainty in position measurement) nontrivially reflects the existence of a minimal measurable length of the order of the Planck length. The existence of such a minimal length is addressed in several approaches to quantum gravity proposal. On the other hand, Doubly Special Relativity (DSR) theories have revealed that due to the existence of a minimal measurable length, there would be also a maximal momentum for a test particle. These are actually two faces of one fact, that is, a natural ultraviolet cutoff which plays the role of a regulator in quantum field theory. A GUP consisting these cutoffs has some important impacts on the foundation of standard quantum mechanics; one of which is the discreteness of the spacetime manifold. In this paper we study Kernel and Feynman Path Integrals in the framework of a GUP that admits a minimal length and a maximal momentum. We work in the quasi-space representation by adopting maximally localized states in position space. We perform our analysis for both the particle-like and wave-like scenarios. We show that unlike the ordinary quantum mechanics, it is possible to construct a path integral even in the wave-like approach due to the presence of natural cutoffs. As an important application, by using the connection between the path integral formalism and statistical mechanics we investigate the modifications imposed on some thermodynamical properties of an ideal gas in this setup.

Generalized uncertainty principle with maximal observable momentum and no minimal length indeterminacy

“…Consequently, we would automatically achieve UV completion from a quantum field theoretical point of view, as p max acts as a regulator, thereby assuring predictability. The very same opinion is commonly encountered in the QFT extensions of GUP; see for instance references [37,38] and therein. • The opportunity to precisely localize a gravitating quantum system is intimately tied to the picture where spacetime has a lattice-like structure which appears as a continuum at ordinary scales.…”